QUADRATIC RELATIONS AND CONIC SECTIONSsciannamath.weebly.com/.../2/...and_conic_sections.pdf · Chapter 10 is about conic sections. The four conic sections are parabolas, circles,

� � � � � � � �QUADRATICRELATIONS ANDCONIC SECTIONS

QUADRATICRELATIONS ANDCONIC SECTIONS

c What shape does a telescopemirror have?

586

� � � � � � � �

APPLICATION: Telescopes

Using powerful telescopes like

the Hubble telescope, astronomers

have discovered planets that exist

outside our solar system. By exam-

ining the wobble in a star’s motion,

astronomers can detect the pres-

ence of one or more planets orbit-

ing that star. The challenge is to

see the planets, but that would

require a telescope with a mirror

100 meters in diameter, 10 times

larger than any existing telescope.

Think & Discuss

The diagram shows a cross section of a mirror from

a telescope at the Big Bear Solar Observatory in

California. An equation for the surface of the mirror,

based on the coordinate system shown, is y = }10

x

4

2

0}

where x and y are measured in centimeters.

1. What shape is the cross section of the mirror?

2. The mirror has a diameter of 65 cm. What is the

depth of the mirror? How did you get your answer?

Learn More About It

In Exercise 67 on p. 630 you will use your knowledge

of conic sections to determine what shape a telescope’s

mirrors have.

APPLICATION LINK Visit www.mcdougallittell.com

for more information about telescopes.

INT

ERNET

C H A P T E R

10

587

c

� �

� � � � � � � �

588 Chapter 10

What’s the chapter about?

Chapter 10 is about conic sections. The four conic sections are parabolas, circles,ellipses, and hyperbolas. In Chapter 10 you’ll learn

• how to use the distance and midpoint formulas.

• how to graph and write equations of conics, and how to classify conics.

• how to solve systems of quadratic equations.

CHAPTER

10Study Guide

PREVIEW

Are you ready for the chapter?

SKILL REVIEW Do these exercises to review key skills that you’ll apply in this chapter.See the given reference page if there is something you don’t understand.

Write an equation of the line that passes through the given point and has the

given slope. (Review Example 2, p. 92)

1. (0, 4), m = 2 2. (2, º2), m = }

13

} 3. (º4, 1), m = º}

43

}

Solve the system using any algebraic method. (Review Examples 1–3, pp. 148–150)

4. x + 2y = 8 5. 2x + y = 3 6. 4x º y = 73x º y = 3 3x + y = 2 5x º 2y = 2

Graph the function. Label the vertex and axis of symmetry.

(Review Examples 1–3, pp. 250 and 251)

7. y = x2 + 4 8. y = º3x2 9. y = 2(x º 3)2 º 1

Solve the equation by completing the square. (Review Examples 2 and 3, p. 283)

10. x2 + 8x + 14 = 0 11. 5x

2 + 15x = º25 12. x2 º 2x = º8x + 14

PREPARE

Here’s a study strategy!

STUDY

STRATEGY

c Review

• parabola, p. 249

• hyperbola, p. 540

c New

• distance formula, p. 589

• midpoint formula, p. 590

• circle, p. 601

• ellipse, p. 609


• conic sections, p. 623

• general second-degreeequation, p. 626

• discriminant, p. 626

KEY VOCABULARY

STUDENT HELP

Study Tip“Student Help” boxesthroughout the chaptergive you study tips andtell you where to look forextra help in this bookand on the Internet.

Dictionary of GraphsMake a dictionary of graphs to use as a referencetool. Draw and label an example of each conic. Notethe important characteristics of the conic, and writethe conic’s equation. Expand your dictionary toinclude all the types of graphs you have learned andcontinue to learn in this course.

� � � � � � �

10.1 The Distance and Midpoint Formulas 589

The Distance and Midpoint Formulas

USING THE DISTANCE AND MIDPOINT FORMULAS

To find the distance d between A(x1, y1) and B(x2, y2), youcan apply the Pythagorean theorem to right triangle ABC.

(AB)2 = (AC)2 + (BC)2

d2 = (x2 º x1)2 + (y2 º y1)2

d = Ï(xw2wºw xw1)w2w+w (wy2w ºw yw1)w2wThe third equation is called the

Finding the Distance Between Two Points

Find the distance between (º2, 5) and (3, º1).

SOLUTION

Let (x1, y1) = (º2, 5) and (x2, y2) = (3, º1).

d = Ï(xw2wºw xw1)w2w+w (wy2w ºw yw1)w2w Use distance formula.

= Ï(3w ºw (wºw2w))w2w+w (wºw1w ºw 5w)2w Substitute.

= Ï2w5w +w 3w6w Simplify.

= Ï6w1w ≈ 7.81 Use a calculator.

Classifying a Triangle Using the Distance Formula

Classify ¤ABC as scalene, isosceles, or equilateral.

SOLUTION

AB = Ï(6w ºw 4w)2w +w (w1w ºw 6w)2w = Ï2w9w

BC = Ï(1w ºw 6w)2w +w (w3w ºw 1w)2w = Ï2w9w

AC = Ï(1w ºw 4w)2w +w (w3w ºw 6w)2w = 3Ï2w

c Because AB = BC, ¤ABC is isosceles.

E X A M P L E 2

E X A M P L E 1

distance formula.

GOAL 1

RE

AL LIFE

RE

AL LIFE

Find the distance

between two points and find

the midpoint of the line seg-

ment joining two points.

Use the distance

and midpoint formulas in

real-life situations, such as

finding the diameter of a

broken dish in Example 5.

. To solve real-life

problems, such as finding the

distance a medical helicopter

must travel to a hospital in

Exs. 53–56.

Why you should learn it

GOAL 2

GOAL 1

What you should learn

10.1

E X P L O R I N G D ATA

A N D S TAT I S T I C S

The distance d between the points (x1, y1) and (x2, y2) is as follows:

d = Ï(xw2wºw xw1)w2w+w (wyw2wºw yw1)w2w

THE DISTANCE FORMULA

��A(x1, y1) C (x2, y1)

B(x2, y2)

d

�� C

(1, 3)

A (4, 6)

B (6, 1)

� � � � � � � �Another formula involving two points in a coordinate plane is the

Recall that the midpoint of a segment is the point on the segment that is equidistant from the two endpoints.

Finding the Midpoint of a Segment

Find the midpoint of the line segment joining (º7, 1) and (º2, 5).

SOLUTION

Let (x1, y1) = (º7, 1) and (x2, y2) = (º2, 5).

S}x1 +

2

x2}, }

y1 +

2

y2}D = S , }

1 +2

5}D

= Sº}

92

}, 3D

Finding a Perpendicular Bisector

Write an equation for the perpendicular bisector of the line segment joining A(º1, 4) and B(5, 2).

SOLUTION

First find the midpoint of the line segment:

S}x1 +

2

x2}, }

y1 +

2

y2}D = S}º1

2+ 5}, }

4 +2

2}D = (2, 3)

Then find the slope of ABÆ

:

m = }y

x2

2

º

º

y

x1

1} = }

52º

º(º

41)

} = }º62} = º}

13

}

The slope of the perpendicular bisector is the negative reciprocal of º}

13

}, or mfi = 3.

Since you know the slope of the perpendicular bisector and a point that the bisectorpasses through, you can use the point-slope form to write its equation.

y º 3 = 3(x º 2)

y = 3x º 3

c An equation for the perpendicular bisector of ABÆ

is y = 3x º 3.

E X A M P L E 4

º7 + (º2)}}

2

E X A M P L E 3

formula.midpoint

590 Chapter 10 Quadratic Relations and Conic Sections

The midpoint of the line segment joining

A(x1, y1) and B (x2, y2) is as follows:

MS}x1 +

2

x2}, }

y1 +

2

y2}D

Each coordinate of M is the mean of the

corresponding coordinates of A and B.

THE MIDPOINT FORMULA � �A(x1, y1)

B(x2, y2)

midpoint M

��

2

� �(27, 1)

s2 , 3d92

(22, 5)

� B(5, 2)

��

y 5 3x 2 3

(2, 3)A(21, 4)

Look Back For help withperpendicular lines, see p. 92.

STUDENT HELP

� � � � � � � �DISTANCE AND MIDPOINT FORMULAS IN REAL LIFE

Recall from geometry that the perpendicular bisector of a chord of a circle passesthrough the center of the circle. Using this theorem, you can find the center of acircle given three points on the circle.

Using the Distance and Midpoint Formulas in Real Life

ARCHEOLOGY While on an archeological dig, youdiscover a piece of a broken dish. To estimate theoriginal diameter of the dish, you lay the piece on acoordinate plane and mark three points on thecircular edge, as shown. Use these points to find thediameter of the dish. (Each unit in the coordinateplane represents 1 inch.)

SOLUTION

Use the method illustrated in Example 4 to find theperpendicular bisectors of AO

Æand OB

Æ.

y = 2x + 5 Perpendicular bisector of AOÆ

y = º}

32

}x + }

123} Perpendicular bisector of OB

Æ

Both bisectors pass through the circle’s center.Therefore, the center of the circle is the solution of the system formed by these two equations.

y = 2x + 5 Write first equation.

º}

32

}x + }

123} = 2x + 5 Substitute for y.

º3x + 13 = 4x + 10 Multiply each side by 2.

º7x = º3 Simplify.

x = }

37

} Divide each side by º7.

y = 2S}

37

}D + 5Substitute the x-value

into the first equation.

y = }

471} Simplify.

The center of the circle is CS}

37

} , }

471}D. The radius of the circle is the distance

between C and any of the three given points.

CO = !S§0§ º§ }

37§}D§2§+§ S§0§ º§ }

4§71}§D§2§

= !}

1§46§990}§

≈ 5.87

c The dish had a diameter of about 2(5.87) = 11.74 inches.

E X A M P L E 5

GOAL 2


(24, 2)

!

O (0, 0)

B(6, 4)

! "A

"! !

O

C

B

A

ARCHEOLOGISTS

use grids to system-atically explore a site. Bylabeling the grid squares,they can record where eachartifact is found.

RE

AL LIFE

RE

AL LIFE

FOCUS ON

APPLICATIONS

Look Back For help with solvingsystems, see p. 148.

STUDENT HELP

# $ % & ' ( ) *


1. State the distance and midpoint formulas.

2. Look back at Example 1. Find the distance between (º2, 5) and (3, º1), but thistime letting (x1, y1) = (3, º1) and (x2, y2) = (º2, 5). How are the calculationsdifferent? Do you get the same answer?

3. a. Write a formula for the distance between a point (x, y) and the origin.

b. Write a formula for the midpoint of the segment joining a point (x, y) and the origin.

Find the distance between the two points.

4. (2, º1), (2, 3) 5. (º5, º2), (0, º2) 6. (0, 6), (4, 9)

7. (10, º2), (7, 4) 8. (º3, 8), (5, 6) 9. (6, º1), (º9, 8)

Find the midpoint of the line segment joining the two points.

10. (0, 0), (º8, 14) 11. (0, 3), (4, 9) 12. (1, º2), (1, 6)

13. (1, 3), (3, 11) 14. (º5, 4), (2, º4) 15. (º1, 5), (º8, º6)

16. HIKING You are going on a two-day hike.The map at the right shows the trails you planto follow. (Each unit represents 1 mile.)

a. You hike from the lodge to point A and decidethat you will hike to the midpoint of AB

Æbefore

you camp for the night. At what point in theplane will you be camping?

b. How far will you hike each day?

USING THE FORMULAS Find the distance between the two points. Then find

the midpoint of the line segment joining the two points.

17. (0, 0), (3, 4) 18. (0, 0), (4, 12) 19. (0, 4), (8, º3)

20. (º2, 8), (6, 0) 21. (º3, º1), (7, 4) 22. (9, º2), (3, 6)

23. (º5, º8), (1, 6) 24. (º2, 10), (10, º2) 25. (8, 3), (2, º1)

26. (º10, º15), (12, 18) 27. (º3.5, 1.2), (6, º3.8) 28. (6.3, º9), (1.3, º8.5)

29. (º7, 2), Sº}

121}, 4D 30. S}

23

}, º}

141}D, Sº}

72

}, º}

121}D 31. Sº}

34

}, 2D, S5, º}

74

}DThe vertices of a triangle are given. Classify the

triangle as scalene, isosceles, or equilateral.

32. (2, 0), (0, 8), (º2, 0) 33. (4, 1), (1, º2), (6, º4) 34. (1, 9), (º4, 2), (4, 2)

35. (2, 5), (8, 2), (4, º1) 36. (5, º1), (º4, 0), (3, 5) 37. (4, 4), (8, 1), (6, º5)

38. (0, º3), (3, 5), (º5, 2) 39. (1, 1), (º4, 0), (º2, 5) 40. (2, 4), (3, º2), (º1, 1)

GEOMETRY CONNECTION

PRACTICE AND APPLICATIONS

GUIDED PRACTICE

Vocabulary Check ✓Concept Check ✓

Skill Check ✓

"

lodge (0, 0)

B(2, 6)

A(23, 2)

Extra Practiceto help you masterskills is on p. 953.

STUDENT HELP

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 17–31,47–50

Example 2: Exs. 32–40Example 3: Exs. 17–31Example 4: Exs. 41–46Example 5: Exs. 51–58

# $ % & ' ( ) *FINDING EQUATIONS Write an equation for the perpendicular bisector of the

line segment joining the two points.

41. (2, 2), (6, 14) 42. (0, 0), (º8, º10) 43. (0, º6), (º4, 9)

44. (3, º7), (º3, 1) 45. (º3, º7.2), (º4.2, 1.8) 46. S}

32

}, º6D, (º3, 1)

FINDING A COORDINATE Use the given distance d between the two points to

solve for x.

47. (0, 1), (x, 4); d = Ï3w4w 48. (1, 3), (º6, x); d = Ï7w4w

49. (x, º10), (º8, 4); d = 7Ï5w 50. (0.5, x), (7, 2); d = 8.5

URBAN PLANNING In Exercises 51

and 52, use the following information.

You are designing a city park like the one shown at the right. You want the park to have two fountains so that each fountain is equidistant from four of the six park entrances. The labeled points shown in the coordinate plane represent the park entrances.

51. Where should the fountains be placed?

52. How far apart should the fountains be placed?

HELICOPTER RESCUE In Exercises 53–56,

use the following information to find the

distance a medical helicopter would have

to travel to St. John’s Hospital from each

highway intersection.

The Highway Department of SangamonCounty in Illinois uses a map with acoordinate plane whose origin representsdowntown Springfield. Each unit representsone mile and the letters N, S, E, and W areused to indicate the direction. For example,3E 5S corresponds to (3, º5), a point 3 mileseast and 5 miles south of downtown Springfield.St. John’s Hospital is located at 1E 0, or (1, 0).

53. Rt. 1–Rt. 32 intersection at 19E 6N 54. Rt. 37–Rt. 40 intersection at 6E 9S

55. Rt. 18–Rt. 40 intersection at 6W 9S 56. Rt. 10–Rt. 47 intersection at 14W 1N

57. ACCIDENT RECONSTRUCTION When a car makes a fast, sharp turn, anaccident reconstructionist can use the car’s skid mark to determine its speed. The equation v = Ïawrw gives the car’s speed v (in meters per second) as afunction of the radius r of the circle (in meters) along which the car wastraveling. The constant a (measured in meters per second squared) variesdepending on road conditions. Find the radius of the skid mark shown below.Then use the given equation and 6.86 for a to find how fast the car was going.

c Source: Mathematical Modeling


!! "(210, 1)

(0, 0)(16, 2)

ACCIDENT

RECONSTRUC-

TIONIST An accidentreconstructionist usesphysical evidence, such asskid marks, to determinehow accidents occurred.

CAREER LINK

www.mcdougallittell.com

INT

ERNET

RE

AL LIFE

RE

AL LIFE

FOCUS ON

CAREERS

HOMEWORK HELP

Visit our Web sitewww.mcdougallittell.comfor help with problemsolving in Exs. 51 and 52.

INT

ERNET

STUDENT HELP

1

32

3740

10 18

St. John'sHospital

6E9S

6W9S

1E 0

+,14W

1N19E6N

47

# $ % & ' ( ) *


58. A physician uses many tests to evaluate a patient’scondition. Some of these tests yield numerical results. In these cases, thephysician can treat two test results as an ordered pair and use the distanceformula to determine how close to average the patient is. In the table below the serum creatinine (C) and systolic blood pressure (P) for several patients are given. Tell how far from normal each patient is, where normal is represented by the ordered pair (C, P) = (1, 127).

QUANTITATIVE COMPARISON In Exercises 59º62, choose the statement that

is true about the given quantities.

¡A The quantity in column A is greater.

¡B The quantity in column B is greater.

¡C The two quantities are equal.

¡D The relationship cannot be determined from the given information.

59.

60.

61.

62.

63. FINDING A FORMULA Find formulas for the distance between a point (x, y)and each of the following: (a) a horizontal line y = k and (b) a vertical line x = h.

GRAPHING FUNCTIONS Graph the quadratic function. (Review 5.1 for 10.2)

64. y = 4x2 65. y = 3x2 66. y = º3x2 67. y = º2x2

68. y = }

13

}x2 69. y = º}

23

}x2 70. y = º}

34

}x2 71. y = }

56

}x2

SOLVING EQUATIONS Solve the equation. Check for extraneous solutions.

(Review 7.6)

72. x2/3 + 13 = 17 73. Ïxw+w 1w0w0w = 25 74. Ï2wxw = x º 4

75. Ïxw+w 2w = Ï3wxw 76. 2Ï33wxw = 6 77. º2x3/2 = º8

OPERATIONS WITH RATIONAL EXPRESSIONS Perform the indicated

operation and simplify. (Review 9.5)

78. }

x +2

1} º 79. + }

31x} 80. }

4(x1º1

5)} º }

x4+x

1}

81. º }

xx

º+

13

} 82. }

3x2+ 2} + }

x5ºx2

4} 83. }

1x

ºº

36x

} + }2x

2+ 1}

3x}

x2

4}

2x2

x}

x2 º 1

MIXED REVIEW

STATISTICS CONNECTION

C 2 5 1 7 3 4 1

P 120 127 140 115 112 125 130

★★Challenge

TestPreparation

Column A Column B

Distance between (0, 7) and (1, º1) Distance between (9, 2) and (3, 8)

Distance between (º5, º2) and (5, 2) Distance between (º5, 5) and (2, º2)

Distance between (º3, 0) and (2, º4) Distance between (7, 6) and (1, 5)

Distance between (2, º5) and (1, 6) Distance between (0, 8) and (6, 0)

- . / 0 1 2 3 4

10.2 Parabolas 595

Parabolas

GRAPHING AND WRITING EQUATIONS OF PARABOLAS

You already know that the graph of y = ax2 is a parabola whose vertex (0, 0) lies onits axis of symmetry x = 0. Every parabola has the property that any point on it isequidistant from a point called the and a line called the

In Chapter 5 you saw parabolas that have a vertical axis of symmetry and open up ordown. In this lesson you will also work with parabolas that have a horizontal axis ofsymmetry and open left or right. In the four cases shown below, the focus and thedirectrix each lie |p| units from the vertex.

Characteristics of the parabolas shown above are given on the next page.

+,

y 2 5 4px, p < 0

vertex: (0, 0)

directrix:x 5 2p

(p, 0)

focus:

56

y 2 5 4px, p > 0

vertex: (0, 0)

directrix:x 5 2p

(p, 0)

focus:

5 6x 2 5 4py, p < 0

vertex: (0, 0)

directrix: y 5 2p

focus:(0, p)

5 6x 2 5 4py, p > 0

vertex: (0, 0)

directrix: y 5 2p

focus:(0, p)

directrix.focus

GOAL 1

Graph and write

equations of parabolas.

Use parabolas

to solve real-life problems,

such as finding the depth

of a solar energy collector

in Example 3.

. To model real-lifeparabolas, such as the

reflector of a car’s headlight

in Ex. 79.


GOAL 2

GOAL 1


10.2R

EAL LIFE

REA

L LIFE

The vertex lies halfwaybetween the focus andthe directrix.

The focus lies on theaxis of symmetry.

The directrix isperpendicular to theaxis of symmetry.

7 8 9 : ; < = >


Graphing an Equation of a Parabola

Identify the focus and directrix of the parabola given by x = º}

16}y2. Draw the parabola.

SOLUTION

Because the variable y is squared, the axis of symmetry is horizontal. To find thefocus and directrix, rewrite the equation as follows.

x = º}

16}y2 Write original equation.

º6x = y2 Multiply each side by º6.

Since 4p = º6, you know p = º}

32}. The focus is

(p, 0) = Sº}

32}, 0D and the directrix is x = ºp = }

32}.

To draw the parabola, make a table of values and plot points. Because p < 0, the parabola opens to the left. Therefore, only negative x-values should be chosen.

Writing an Equation of a Parabola

Write an equation of the parabola shown at the right.

SOLUTION

The graph shows that the vertex is (0, 0) and thedirectrix is y = ºp = º2. Substitute 2 for p in thestandard equation for a parabola with a vertical axis of symmetry.

x2 = 4py Standard form, vertical axis of symmetry

x2 = 4(2)y Substitute 2 for p.

x2 = 8y Simplify.

✓CHECK You can check this result by solving the equation for y to get y = }

18}x2

and graphing the equation using a graphing calculator.

E X A M P L E 2

E X A M P L E 1

The standard form of the equation of a parabola with vertex at (0, 0) is as

follows.

EQUATION FOCUS DIRECTRIX AXIS OF SYMMETRY

x2 = 4py (0, p) y = ºp Vertical (x = 0)

y2 = 4px (p, 0) x = ºp Horizontal (y = 0)

STANDARD EQUATION OF A PARABOLA (VERTEX AT ORIGIN)

?? @ Adirectrix

vertex

B @ ? Ax 532

s d32

, 02

x º1 º2 º3 º4 º5

y ±2.45 ±3.46 ±4.24 ±4.90 ±5.48

Look Back For help with drawingparabolas, see p. 249.

STUDENT HELP

HOMEWORK HELP

Visit our Web sitewww.mcdougallittell.comfor extra examples.

INT

ERNET

STUDENT HELP

7 8 9 : ; < = >

10.2 Parabolas 597

USING PARABOLAS IN REAL LIFE

Parabolic reflectors have cross sections that are parabolas.A special property of any parabolic reflector is that allincoming rays parallel to the axis of symmetry that hit thereflector are directed to the focus (Figure 1). Similarly, raysemitted from the focus that hit the reflector are directed inrays parallel to the axis of symmetry (Figure 2). Theseproperties are the reason satellite dishes and flashlights are parabolic.

Modeling a Parabolic Reflector

SOLAR ENERGY Sunfire is a glass parabola usedto collect solar energy. The sun’s rays are reflectedfrom the mirrors toward two boilers located at thefocus of the parabola. When heated, the boilersproduce steam that powers an alternator to produceelectricity.

a. Write an equation for Sunfire’s cross section.

b. How deep is the dish?

SOLUTION

a. The boilers are 10 feet above the vertex of the dish. Because the boilers are at thefocus and the focus is p units from the vertex, you can conclude that p = 10.

Assuming the vertex is at the origin, an equation for the parabolic cross section isas follows:

x2 = 4py Standard form, vertical axis of symmetry

x2 = 4(10)y Substitute 10 for p.

x2 = 40y Simplify.

b. The dish extends }

327} = 18.5 feet on either side of the origin. To find the depth

of the dish, substitute 18.5 for x in the equation from part (a).

x2 = 40y Equation for the cross section

(18.5)2 = 40y Substitute 18.5 for x.

8.6 ≈ y Solve for y.

c The dish is about 8.6 feet deep.

E X A M P L E 3

GOAL 2

HOWARD FRANK

BROYLES is anengineer and inventor. More than 250 high schoolstudents volunteered theirtime to help him buildSunfire. The project tookover ten years.

APPLICATION LINK


INT

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FOCUS ON

PEOPLE

boiler

depth10 ft

37 ft

Figure 1 Figure 2

Sunfire

C D E F G H I J


1. Complete this statement: A parabola is the set of points equidistant from a pointcalled the

ooo

? and a line called the ooo

? .

2. How does the graph of x = ay2 differ from the graph of y = ax2?

3. Knowing the value of a in y = ax2, how can you find the focus and directrix?

Graph the equation. Identify the focus and directrix of the parabola.

4. x2 = 4y 5. y = º5x2 6. º12x = y2

7. 8y2 = x 8. º6x = y2 9. x2 = 2y

Write the standard form of the equation of the parabola with the given focus or

directrix and vertex at (0, 0).

10. focus: (0, 3) 11. focus: (5, 0) 12. focus: (º6, 0)

13. directrix: x = 4 14. directrix: x = º1 15. directrix: y = 8

MATCHING Match the equation with its graph.

16. y2 = 4x 17. x2 = º4y 18. x2 = 4y

19. y2 = º4x 20. y2 = }

14}x 21. x2 = }

14}y

A. B. C.

D. E. F.

DIRECTION Tell whether the parabola opens up, down, left, or right.

22. y = º3x2 23. º9x2 = 2y 24. 2y2 = º6x 25. x = 7y2

26. x2 = 16y 27. º3y2 = 8x 28. º5x = ºy2 29. x2 = }

43}y

FOCUS AND DIRECTRIX Identify the focus and directrix of the parabola.

30. 3x2 = ºy 31. 2y2 = x 32. x2 = 8y 33. y2 = º10x

34. y2 = º16x 35. x2 = º36y 36. º4x + 9y2 = 0 37. º28y + x2 = 0

K L2 M MK M LNK MM L

K M L2 MKM LNK MM L


GUIDED PRACTICE

Vocabulary Check ✓

Concept Check ✓

Skill Check ✓


STUDENT HELP

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 16–53Example 2: Exs. 54–77Example 3: Exs. 78–81

O P Q R S T U V

10.2 Parabolas 599

GRAPHING Graph the equation. Identify the focus and directrix of the parabola.

38. y2 = 12x 39. x2 = º6y 40. y2 = º2x 41. y2 = 24x

42. x2 = 8y 43. y2 = º14x 44. x2 = º20y 45. x2 = 18y

46. x2 = º4y 47. x2 = 16y 48. y2 = 9x 49. y2 = º3x

50. x2 º 40y = 0 51. x + }210}y2 = 0 52. 3x2 = 4y 53. x º }

18}y2 = 0

WRITING EQUATIONS Write the standard form of the equation of the parabola

with the given focus and vertex at (0, 0).

54. (4, 0) 55. (º2, 0) 56. (º3, 0) 57. (0, 1)

58. (0, 4) 59. (0, º3) 60. (0, º4) 61. (º5, 0)

62. Sº}

14}, 0D 63. S0, º}

38}D 64. S0, }

12}D 65. S}1

52}, 0D

WRITING EQUATIONS Write the standard form of the equation of the parabola

with the given directrix and vertex at (0, 0).

66. y = 2 67. y = º3 68. x = º4 69. x = 6

70. x = º5 71. y = º1 72. x = 2 73. y = 4

74. x = º}

12} 75. x = }

34} 76. y = }

58} 77. y = º}1

12}

78. COMMUNICATIONS The cross section of a television antenna dish is a parabola. For the dish at theright, the receiver is located at the focus, 4 feet above the vertex. Find an equation for the cross section of thedish. (Assume the vertex is at the origin.) If the dish is 8 feet wide, how deep is it?

79. AUTOMOTIVE ENGINEERING The filament of alightbulb is a thin wire that glows when electricity passesthrough it. The filament of a car headlight is at the focusof a parabolic reflector, which sends light out in a straightbeam. Given that the filament is 1.5 inches from thevertex, find an equation for the cross section of thereflector. If the reflector is 7 inches wide, how deep is it?

80. In the drawing shown at the left, the rays of the sun arelighting a candle. If the candle flame is 12 inches from the back of the parabolicreflector and the reflector is 6 inches deep, then what is the diameter of thereflector?

81. CAMPING You can make a solar hot dog cooker using foil-lined cardboard shaped as aparabolic trough. The drawing at the right showshow to suspend a hot dog with a wire throughthe focus of each end piece. If the trough is 12 inches wide and 4 inches deep, how farfrom the bottom should the wire be placed? c Source: Boys’ Life

12 in.

4 in.

HISTORY CONNECTION

4 ft

8 ft

MATH IN HISTORY

Man has beenaware of the reflectiveproperties of parabolas for two thousand years. The above illustration ofArchimedes’ burning mirrorwas taken from a bookprinted in the 17th century.

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7 in.

1.5 in.

O P Q R S T U V


82. Writing For an equation of the form y = ax2, discuss what effect increasing|a| has on the focus and directrix.

83. MULTI-STEP PROBLEM A flashlight has a parabolic reflector. An equation for

the cross section of the reflector is y2 = }

372}x. The depth of the reflector is }

32} inches.

a. Writing Explain why the value of p must be less than the depth of the reflector of a flashlight.

b. How wide is the beam of light projected by the flashlight?

c. Write an equation for the cross section of a reflector having the same depth but a wider beam than the flashlight shown. How wide is the beam of the new reflector?

d. Write an equation for the cross section of a reflector having the same depthbut a narrower beam than the flashlight shown. How wide is the beam of thenew reflector?

84. The latus rectum

of a parabola is the line segment that isparallel to the directrix, passes through thefocus, and has endpoints that lie on theparabola. Find the length of the latusrectum of a parabola given by x2 = 4py.

LOGARITHMIC AND EXPONENTIAL EQUATIONS Solve the equation. Check for

extraneous solutions. (Review 8.6)

85. 85x = 162x + 1 86. 3x = 15 87. 5x = 7

88. 103x + 1 + 4 = 33 89. log7 (3x º 5) = log7 8x 90. log3 (4x º 3) = 3

OPERATIONS WITH RATIONAL EXPRESSIONS Perform the indicated

operation and simplify. (Review 9.4 and 9.5)

91. }

3

x

x

3

y

y

3

} • }6y

x} 92. }

32x

x

y3

} ÷ }23x

x

y3

} 93. }x2

x

º

2 ºx º

96

} • (x + 2)

94. }

xº+3x

2} + }x

4ºx

1} 95. }

x

6+x2

1} º }

6x

x2++

16x

} 96. }x2 º

x º3x

1+ 2

} º

FINDING A DISTANCE Find the distance between the two points.

(Review 10.1 for 10.3)

97. (3, 4), (6, 7) 98. (º3, 7), (º7, 3) 99. (18, º4), (º2, 9)

100. (3.7, 5.1), (2, 5) 101. (º9, º31), (8, 7) 102. (8.8, 3.3), (1.2, 6)

103. CONSUMER ECONOMICS The amount A (in dollars) you pay for grapesvaries directly with the amount P (in pounds) that you buy. Suppose you buy

1}

12} pounds for $2.25. Write a linear model that gives A as a function of P.

(Review 2.4)

x2 º 4}

x º 2

MIXED REVIEW

GEOMETRY CONNECTION

TestPreparation

★★Challenge

EXTRA CHALLENGE


W Xy 5 2p

latusrectum

x 2 5 4py

bulb

reflector

filament

Y Z [ \ ] ^ _ `

10.3 Circles 601

Circles

GRAPHING AND WRITING EQUATIONS OF CIRCLES

A is the set of all points (x, y) that areequidistant from a fixed point, called the ofthe circle. The distance r between the center and anypoint (x, y) on the circle is the

The distance formula can be used to obtain anequation of the circle whose center is the origin andwhose radius is r. Because the distance between anypoint (x, y) on the circle and the center (0, 0) is r,you can write the following.

Ï(xw ºw 0w)2w +w (wywºw 0w)2w = r Distance formula

(x º 0)2 + (y º 0)2 = r2 Square both sides.

x2 + y2 = r2 Simplify.

Graphing an Equation of a Circle

Draw the circle given by y2 = 25 º x2.

SOLUTION

Write the equation in standard form.

y2 = 25 º x2 Original equation

x2 + y2 = 25 Add x2 to each side.

In this form you can see that the graph is a circlewhose center is the origin and whose radius isr = Ï2w5w = 5.

Plot several points that are 5 units from theorigin. The points (0, 5), (5, 0), (0, º5), and(º5, 0) are most convenient.

Draw a circle that passes through the four points.

E X A M P L E 1

radius.

center

circle

GOAL 1

Graph and write

equations of circles.

Use circles to

solve real-life problems,

such as determining whether

you are affected by an

earthquake in Ex. 81.

. To model real-lifesituations with circular

models, such as the region

lit by a lighthouse beam in

Example 4.


GOAL 2

GOAL 1


10.3

The with center at (0, 0) and

radius r is as follows:

x2 + y2 = r2

EXAMPLE A circle with center at (0, 0) and radius 3 has equation x2 + y2 = 9.

standard form of the equation of a circle

STANDARD EQUATION OF A CIRCLE (CENTER AT ORIGIN)

ab(x, y)

r

x2 1 y2 5 r 2

cc d(0, 5)

e(5, 0)(25, 0)

(0, 25)

y 2 5 25 2 x2

REA

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f g h i j k l m


Writing an Equation of a Circle

The point (1, 4) is on a circle whose center is the origin. Write the standard form ofthe equation of the circle.

SOLUTION

Because the point (1, 4) is on the circle, the radius of the circle must be the distancebetween the center and the point (1, 4).

r = Ï(1w ºw 0w)2w +w (w4w ºw 0w)2w Use the distance formula.

= Ï1w +w 1w6w Simplify.

= Ï1w7w

Knowing that the radius is Ï1w7w, you can use the standard form to find an equation ofthe circle.

x2 + y2 = r2 Standard form

x2 + y2 = (Ï1w7w)2Substitute Ï1w7w for r.

x2 + y2 = 17 Simplify.

. . . . . . . . . .

A theorem in geometry states that a line tangent toa circle is perpendicular to the circle’s radius at thepoint of tangency. In the diagram, AB@#$ is tangent tothe circle with center C at the point of tangency B,so AB@#$ fi BwCw. This property of circles is used in thenext example.

Finding a Tangent Line

Write an equation of the line that is tangent to the circle x2 + y2 = 13 at (2, 3).

SOLUTION

The slope of the radius through the point (2, 3) is:

m = }32ºº

00} = }2

3}

Because the tangent line at (2, 3) is perpendicularto this radius, its slope must be the negative

reciprocal of }32}, or º}

23}. So, an equation of the

tangent line is as follows.

y º 3 = º}23}(x º 2) Point-slope form

y º 3 = º}23}x + }3

4} Distributive property

y = º}23}x + }

133} Add 3 to each side.

c An equation of the tangent line is y = º}23}x + }

133}.

E X A M P L E 3

E X A M P L E 2

STUDENT HELP

Study TipIn mathematics the termradius is used in twoways. As defined on theprevious page, it is thedistance from the centerof a circle to a point onthe circle. It can alsorefer to the line segmentthat connects the centerto a point on the circle.

A B

C

HOMEWORK HELP


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STUDENT HELP

nn op(2, 3)

y 5 2 x 123

133

x2 1 y 2 5 13

q r s t u v w x

10.3 Circles 603

USING CIRCLES IN REAL LIFE

The regions inside and outside the circle x2 + y2 = r2

can be described by inequalities.

Region inside circle: x2 + y2 < r2

Region outside circle: x2 + y2 > r2

Using a Circular Model

OCEAN NAVIGATION The beam of a lighthouse can be seen for up to 20 miles. You are on a ship that is 10 miles east and 16 miles north of the lighthouse.

a. Write an inequality to describe the region lit by the lighthouse beam.

b. Can you see the lighthouse beam?

SOLUTION

a. As shown at the right the lighthouse beam can beseen from all points that satisfy this inequality:

x2 + y2 < 202

b. Substitute the coordinates of the ship into theinequality you wrote in part (a).

x2 + y2 < 202 Inequality from part (a)

102 + 162 <? 202 Substitute for x and y.

100 + 256 <? 400 Simplify.

356 < 400 ✓ The inequality is true.

c You can see the beam from the ship.

Using a Circular Model

OCEAN NAVIGATION Your ship in Example 4 is traveling due south. For howmany more miles will you be able to see the beam?

SOLUTION

When the ship exits the region lit by the beam, it will be at a point on the circle x2 + y2 = 202.Furthermore, its x-coordinate will be 10 and its y-coordinate will be negative. Find the point(10, y) where y < 0 on the circle x2 + y2 = 202.

x2 + y2 = 202 Equation for the boundary

102 + y2 = 202 Substitute 10 for x.

y =6Ï3w0w0w ≈617.3 Solve for y.

c Since y < 0, y ≈ º17.3. The beam will be in view as the ship travels from (10, 16) to (10, º17.3), a distance of |16 º (º17.3)| = 33.3 miles.

E X A M P L E 5

E X A M P L E 4

GOAL 2 o p

yy op(10, 16)

x2 1 y2 , 202

THE PHAROS OF

ALEXANDRIA

was a lighthouse built inEgypt in about 280 B.C. Oneof the Seven Wonders of theWorld, it was said to be over440 feet tall. It stood fornearly 1400 years.

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In the diagram above,

the origin represents the

lighthouse and the positive

y-axis represents north.

yy op

(10, 217.3)

q r s t u v w x


1. State the definition of a circle.

2. LOGICAL REASONING Tell whether the following statement is always true,sometimes true, or never true: For a given circle and a given x-coordinate, thereare two points on the circle with that x-coordinate.

3. How is the slope of a line tangent to a circle related to the slope of the radius atthe point of tangency?

4. ERROR ANALYSIS A student was asked to write an equation of a circle with itscenter at the origin and a radius of 4. The student wrote the following equation:

x2 + y2 = 4

What did the student do wrong? Write the correct equation.

Write the standard form of the equation of the circle that passes through the

given point and whose center is the origin.

5. (4, 0) 6. (0, º2) 7. (º8, 6) 8. (º5, º12)

9. (6, º9) 10. (3, 1) 11. (º5, º5) 12. (º2, 4)

Graph the equation. Give the radius of the circle.

13. x2 + y2 = 36 14. x2 + y2 = 81 15. x2 + y2 = 32

16. x2 + y2 = 12 17. 36x2 + 36y2 = 144 18. 9x2 + 9y2 = 162

19. SHOT PUT A person throws a shot put from a circle that has a diameter of7 feet. Write the standard form of the equation of the shot put circle if the centeris the origin.

MATCHING GRAPHS Match the equation with its graph.

20. x2 + y2 = 16 21. x2 + y2 = 5 22. x2 + y2 = 4

23. x2 + y2 = 25 24. x2 + y2 = 100 25. x2 + y2 = 10

A. B. C.

D. E. F. zz { |}} { |zz { |~~ { |~~ { |zz { |


GUIDED PRACTICE


Skill Check ✓


STUDENT HELP

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 20–46Example 2: Exs. 47–70Example 3: Exs. 71–79Examples 4, 5: Exs. 81–87

� � � � � � � �

10.3 Circles 605

GRAPHING Graph the equation. Give the radius of the circle.

26. x2 + y2 = 1 27. x2 + y2 = 49 28. x2 + y2 = 64

29. x2 + y2 = 20 30. x2 + y2 = 8 31. x2 + y2 = 10

32. x2 + y2 = 3 33. 5x2 + 5y2 = 80 34. 24x2 + 24y2 = 96

35. 8x2 + 8y2 = 192 36. 9x2 + 9y2 = 135 37. 4x2 + 4y2 = 52

GRAPHING In Exercises 38–46, the equations of both circles and parabolas are

given. Graph the equation.

38. x2 + y2 = 11 39. x2 + y2 = 1 40. x2 + y = 0

41. }14}x2 + }

14}y2 = 16 42. 4x2 + y = 0 43. 9x2 + 9y2 = 441

44. º2x + 9y2 = 0 45. }38}x2 + }

38}y2 = 6 46. x2 + 12y = 0

WRITING EQUATIONS Write the standard form of the equation of the circle

with the given radius and whose center is the origin.

47. 3 48. 9 49. 6 50. 11

51. Ï7w 52. Ï3w0w 53. Ï1w1w 54. Ï2w1w

55. 5Ï6w 56. 4Ï5w 57. 2Ï7w 58. 3Ï3w

WRITING EQUATIONS Write the standard form of the equation of the circle

that passes through the given point and whose center is the origin.

59. (0, º10) 60. (8, 0) 61. (º3, º4) 62. (º4, º1)

63. (5, º3) 64. (º6, 4) 65. (º6, 1) 66. (º1, º9)

67. (7, º4) 68. (10, 2) 69. (5, 8) 70. (2, º12)

FINDING TANGENT LINES The equation of a circle and a point on the circle is

given. Write an equation of the line that is tangent to the circle at that point.

71. x2 + y2 = 10; (1, 3) 72. x2 + y2 = 5; (2, 1)

73. x2 + y2 = 41; (º4, º5) 74. x2 + y2 = 145; (12, 1)

75. x2 + y2 = 65; (º8, 1) 76. x2 + y2 = 40; (º2, 6)

77. x2 + y2 = 244; (º10, º12) 78. x2 + y2 = }254

7}; S}

12}, º8D

79. CRITICAL THINKING Look back at Example 3. Find an equation of the line that is tangent to the circle at the point (2, º3). Describe how the line isgeometrically related to the line found in Example 3.

80. Writing Describe how the equation of a circle is related to the Pythagoreantheorem. Include a diagram to illustrate the relationship.

81. EARTHQUAKES Suppose an earthquake can be felt up to 80 miles from itsepicenter. You are located at a point 60 miles west and 45 miles south of theepicenter. Do you feel the earthquake? If so, how many miles south would youhave to travel to be out of the range of the earthquake?

82. DESERT IRRIGATION A circular field has an area of about 2,400,000square yards. Write an equation that represents the boundary of the field. Let (0, 0) represent the center of the field.

IRRIGATION Thisirrigation project in

Colorado enables farmers toraise crops in the desert.Water from deep wells ispumped to sprinklers thatrotate, forming circularpatterns.

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83. RESIZING A RING One way to resize a ring is to fit a bar into the ring, as shown. Suppose a ring that is 20 millimeters in diameter has to be resized to fit a finger 16 millimeters in diameter. What is the length of the bar that should be inserted in order to make the ring fit the finger? (Hint: Write an equation of the ring, assuming it is centered at the origin. Determine what the y-coordinate of the bar must be and then substitute this coordinate into the equation to find x.)

84. LIFEGUARD You are a lifeguard at a pond. The pond is a circle with adiameter of 360 feet. You want to rope off a section of the pond for swimming.If you want the rope to form a chord of the circle and have a maximum distanceof 45 feet from shore, approximately how long will you need the rope to be?

85. PHYSICAL THERAPY A tilt-board is a physical therapy device that a person rocks back and forth on. Suppose the ends of a tilt-board are part of a circle with a radius of 30 inches. If the tilt-board has a depth of 6 inches, how wide is it?c Source: Steps to Follow

AIR TRAFFIC CONTROL In Exercises 86 and 87, use the following information.

An air traffic control tower can detect airplanes up to 50 miles away. You are in anairplane 42 miles east and 43 miles south of the control tower.

86. Write an inequality that describes the region in which planes can be detected bythe control tower. Can the control tower detect your plane on its radar?

87. Suppose a jet is 35 miles west and 66 miles north of the control tower and istraveling due south at a speed of 500 miles per hour. After how many minuteswill the jet appear on the control tower’s radar?

88. MULTIPLE CHOICE What is the equation of the line that is tangent to the circlex2 + y2 = 53 at the point (7, 2)?

¡A y = º}72}x + }

425} ¡B y = º}

72}x + }

523}

¡C y = }72}x º }

425} ¡D y = º}

72}x º }

425}

89. MULTIPLE CHOICE Suppose a signal from a television transmitter tower can bereceived up to 150 miles away. The following points represent the locations ofhouses near the transmitter tower with the origin representing the tower. Whichpoint is not within the range of the tower?

¡A (120, 20) ¡B (40, 140) ¡C (105, 120) ¡D (10, 148)

90. ESTIMATING AREA The segment of a circleis the region bounded by a chord and an arc.Estimate the area of the shaded segment byfinding the area of ¤ABC and the area of¤ABD, given that AD

Æand BD

Æare tangent to

the circle.

bar

� �finger

(2x, y) (x, y)

AIR TRAFFIC

CONTROLLER

Air traffic controllers areresponsible for making sureairplanes fly a safe distanceaway from one another. Theyalso help keep flights onschedule.

CAREER LINK


INT

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TestPreparation

★★Challenge

30 in.

6 in.

FOCUS ON

CAREERS

� ��C

D

B(3, 24)

A(3, 4)

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10.3 Circles 607

SOLVING SYSTEMS Solve the system using any algebraic method.

(Review 3.2)

91. x º 9y = 25 92. 9x º y = 8 93. 2x º 3y = 26x º 5y = 3 3x + 10y = º49 º7x + 4y = 6

94. 8x º 5y = 4 95. ºx + 5y = 3 96. º9x + 4y = 152x + y = 1 4x º 9y = 10 3x + 2y = 5

COMPOSITION OF FUNCTIONS Find ƒ(g(x)) and g(ƒ(x)). (Review 7.3)

97. ƒ(x) = x + 1 and g(x) = 2x 98. ƒ(x) = 4x + 1 and g(x) = x º 5

99. ƒ(x) = ºx2 º 1 and g(x) = x + 5 100. ƒ(x) = x2 º 7 and g(x) = 3x + 1

GRAPHING FUNCTIONS Graph the function. (Review 8.1)

101. y = }14} • 5x 102. y = ºS}

53}Dx

103. y = 4 • 3x º 1 º 7

104. y = 3 • 2x º 4 105. y = 2x + 3 º 1 106. y = }14} • 8x + 1

BABYSITTING In Exercises 107 and 108, use the following information.

In June you babysit 35 hours for the Johnsons and 52 hours for the Martins. In Julyyou babysit 112 hours for the Johnsons and 40 hours for the Martins. In August youbabysit 95 hours for the Johnsons and 63 hours for the Martins. (Review 4.1)

107. Use a matrix to organize the information.

108. You charge $6 per hour for babysitting. Using your matrix from Exercise 107,write a matrix that shows how much you earned over the summer vacation.

Find the distance between the two points. Then find the midpoint of the line

segment joining the two points. (Lesson 10.1)

1. (0, 0), (8, 6) 2. (3, 3), (º3, º3) 3. (º2, 7), (4, º10)

4. (3, º7), (º5, º9) 5. (8, 6), (º4, 4) 6. (º1, º13), (11, 15)

Draw the parabola. Identify the focus and directrix. (Lesson 10.2)

7. y2 = 6x 8. 3y = x2 9. ºx2 = 5y 10. º4y2 = 6x

11. 3x2 = 7y 12. }12}x = 2y2 13. x + }

18}y2 = 0 14. ºx2 º 12y = 0

Write the standard form of the equation of the circle that passes through the

given point and whose center is the origin. (Lesson 10.3)

15. (0, 3) 16. (º5, 0) 17. (4, 7) 18. (º2, º5)

19. (º1, 9) 20. (6, º3) 21. (6, º6) 22. (º7, 8)

23. RADIO SIGNALS The signals of a radio station can be received up to 65 miles away. Your house is 35 miles east and 56 miles south of the radiostation. Can you receive the radio station’s signals? Explain. (Lesson 10.3)

QUIZ 1 Self-Test for Lessons 10.1–10.3

MIXED REVIEW

� � � � � � � �Graphing Circles

When you use a graphing calculator to draw a circle, you need to remember

two things. First, most graphing calculators cannot directly graph equations

such as x 2 + y 2 = 36 because they are not functions. Second, to obtain a graph

with true perspective (in which a circle looks like a circle) you must use a

“square setting.”

c EXAMPLE

Use a graphing calculator to draw the graph of x2 + y2 = 36.

c SOLUTION

Begin by solving the equation for y.

x2 + y2 = 36

y2 = 36 º x2

y = ±Ï3w6w ºw xw2w

Enter the two equations into the graphing calculator.

Next set the viewing window so that it has a “square

setting.” On some graphing calculators you can select

a square setting, such as “ZSquare.” On a graphing

calculator whose viewing window’s height is two

thirds its width, you can obtain a “square setting” by

choosing maximum and minimum values that satisfy

this equation:

}

YX

mm

aaxx

ºº

YX

mm

iinn

} = }

23

}

The graph is shown at the right. (Some calculators

may not connect the ends of the two graphs.)

c EXERCISES

Use a graphing calculator to graph the equation. Write the setting of the

viewing window that you used and verify that it is a square setting.

1. x2 + y2 = 121 2. x

2 + y2 = 50 3. x2 + y2 = 484

4. 5x2 + 5y

2 = 120 5. x2 + y2 = }

196} 6. }

12

}x2 + }

12

}y2 = 72

7. }

45

}x2 + }

45

}y2 = 20 8. 9x

2 + 9y2 = 4 9. 125x

2 + 125y2 = 1000

3

2

1


Using Technology

Graphing Calculator Activity for use with Lesson 10.3ACTIVITY 10.3

STUDENT HELP

KEYSTROKE

HELP

See keystrokes for several models ofcalculators atwww.mcdougallittell.com

INT

ERNET

Y1= (36-X2)

Y2=- (36-X2)

Y3=

Y4=

Y5=

Y6=

Y7=

RANGE

Xmin=-12

Xmax=12

Xscl=1

Ymin=-8

Ymax=8

Yscl=1

� � � � � � � �

10.4 Ellipses 609

Ellipses

GRAPHING AND WRITING EQUATIONS OF ELLIPSES

An is the set of all points P such thatthe sum of the distances between P and twodistinct fixed points, called the is aconstant.

The line through the foci intersects the ellipseat two points, the The line segmentjoining the vertices is the and its midpoint is the of the ellipse. The line perpendicular to the major axis at thecenter intersects the ellipse at two points called the The line segmentthat joins these points is the of the ellipse. The two types of ellipses we will discuss are those with a horizontal major axis and those with a vertical major axis.

minor axis

co-vertices.center

major axis,vertices.

foci,

ellipse

GOAL 1

Graph and write

equations of ellipses.

Use ellipses in

real-life situations, such as

modeling the orbit of Mars

in Example 4.

. To solve real-life

problems, such as finding

the area of an elliptical

Australian football field in

Exs. 73 –75.


GOAL 2

GOAL 1


10.4

d1 1 d2 5 constant

d1d2

P

focusfocus

The with center at (0, 0) and major

and minor axes of lengths 2a and 2b, where a > b > 0, is as follows.

EQUATION MAJOR AXIS VERTICES CO-VERTICES

+ = 1 Horizontal (±a, 0) (0, ±b)

+ = 1 Vertical (0, ±a) (±b, 0)

The foci of the ellipse lie on the major axis, c units from the center where

c2 = a2 º b2.

y2

}

a2

x2

}

b2

y2

}

b2

x2

}

a2

standard form of the equation of an ellipse

CHARACTERISTICS OF AN ELLIPSE (CENTER AT ORIGIN)

�co-vertex:(0, b)

co-vertex:(0, 2b)

vertex:(a, 0)

vertex:(2a, 0)

focus:(2c, 0)

focus:(c, 0)

majoraxis

minoraxis

� ��co-vertex:

(b, 0)co-vertex:

(2b, 0)

vertex: (0, a)

vertex: (0, 2a)

focus:(0, 2c)

focus:(0, c)

majoraxis

minoraxis

REA

L LIFE

REA

L LIFE

Ellipse with horizontal major axis

+ = 1y 2

}

b 2

x 2

}

a2

Ellipse with vertical major axis

+ = 1y 2

}

a2

x 2

}

b2

� � � � ¡ ¢ £


Graphing an Equation of an Ellipse

Draw the ellipse given by 9x2 + 16y2 = 144. Identify the foci.

SOLUTION

First rewrite the equation in standard form.

}

194x4

2

} + }

1

1

6

4

y

4

2

} = }

11

44

44

} Divide each side by 144.

}

1x6

2

} + }

y

9

2

} = 1 Simplify.

Because the denominator of the x2-term is greater than that of the y2-term, the major axis is horizontal. So, a = 4 and b = 3. Plot the vertices and co-vertices. Thendraw the ellipse that passes through these four points.

The foci are at (c, 0) and (ºc, 0). To find the valueof c, use the equation c2 = a

2 º b2.

c2 = 42 º 32 = 16 º 9 = 7

c = Ï7w

c The foci are at (Ï7w, 0) and (ºÏ7w, 0).

Writing Equations of Ellipses

Write an equation of the ellipse with the given characteristics and center at (0, 0).

a. Vertex: (0, 7) b. Vertex: (º4, 0)Co-vertex: (º6, 0) Focus: (2, 0)

SOLUTION

In each case, you may wish to draw theellipse so that you have something to checkyour final equation against.

a. Because the vertex is on the y-axis and the co-vertex is on the x-axis, the major axis is vertical with a = 7 and b = 6.

c An equation is }

3x6

2

} + }

4

y

9

2

} = 1.

b. Because the vertex and focus are pointson a horizontal line, the major axis ishorizontal with a = 4 and c = 2. Tofind b, use the equation c

2 = a2 º b2.

22 = 42 º b2

b2 = 16 º 4 = 12

b = 2Ï3w

c An equation is }

1x6

2

} + }

1

y

2

2

} = 1.

E X A M P L E 2

E X A M P L E 1

� �¤¤ (0, 3)

(4, 0)

(0, 23)

(24, 0)

��¥¥vertex (0, 7)

vertex (0, 27)

co-vertex(26, 0)

co-vertex(6, 0)

�¤2

¤vertex(24, 0)

vertex(4, 0)

focus(22, 0)

focus(2, 0)

�

HOMEWORK HELP


INT

ERNET

STUDENT HELP

� � � � ¡ ¢ £

10.4 Ellipses 611

USING ELLIPSES IN REAL LIFE

Both man-made objects, such as The Ellipse at the White House, and naturalphenomena, such as the orbits of planets, involve ellipses.

Finding the Area of an Ellipse

A portion of the White House lawn is called The Ellipse. It is 1060 feet long and 890 feet wide.

a. Write an equation of The Ellipse.

b. The area of an ellipse is A = πab. What is thearea of The Ellipse at the White House?

SOLUTION

a. The major axis is horizontal with

a = }

10260} = 530 and b = }

892

0} = 445.

c An equation is }53

x2

02} + }

44

y2

52} = 1.

b. The area is A = π(530)(445) ≈ 741,000 square feet.

Modeling with an Ellipse

In its elliptical orbit, Mercury ranges from 46.04 million kilometers to 69.86 millionkilometers from the center of the sun. The center of the sun is a focus of the orbit.Write an equation of the orbit.

SOLUTION

Using the diagram shown, you can write a system of linear equations involving a and c.

a º c = 46.04

a + c = 69.86

Adding the two equations gives2a = 115.9, so a = 57.95. Substitutingthis a-value into the second equationgives 57.95 + c = 69.86, so c = 11.91.

From the relationship c2 = a2 º b2, you can conclude the following:

b = Ïaw2wºw cw2w

= Ï(5w7w.9w5w)2w ºw (w1w1w.9w1w)2w

≈ 56.71

c An equation of the elliptical orbit is }(57

x

.9

2

5)2} + }

(56

y

.7

2

1)2} = 1 where x and y are

in millions of kilometers.

E X A M P L E 4

E X A M P L E 3

GOAL 2

RE

AL LIFE

RE

AL LIFE

Landscaping

RE

AL LIFE

RE

AL LIFEAstronomy

Old ExecutiveOffice Building

TreasuryDepartment

White House

The Ellipse

¦§69.86 46.04 ¨ ©ª ©

sun

c

a a

« ¬ ® ¯ ° ± ²


1. Complete each statement using the ellipse shown.

a. The points (º5, 0) and (5, 0) are called the ooo

? .

b. The points (0, º4) and (0, 4) are called the ooo

? .

c. The points (º3, 0) and (3, 0) are called the ooo

? .

d. The segment with endpoints (º5, 0) and (5, 0) iscalled the

ooo

? .

2. How can you tell from the equation of an ellipse whether the major axis is horizontal or vertical?

3. Explain how to find the foci of an ellipse given thecoordinates of its vertices and co-vertices.

4. ERROR ANALYSIS A student was asked to write an equation of the ellipse shown at the right.

The student wrote the equation }

x4

2

} + }

y

9

2

} = 1.

What did the student do wrong? What is the correct equation?


5. Vertex: (0, 5) 6. Vertex: (9, 0) 7. Vertex: (º7, 0)

Co-vertex: (º4, 0) Co-vertex: (0, 2) Focus: (º2Ï1w0w, 0)

8. Vertex: (0, 13) 9. Co-vertex: (Ï9w1w, 0) 10. Co-vertex: (0, Ï3w3w)

Focus: (0, º5) Focus: (0, 3) Focus: (4, 0)

Draw the ellipse.

11. }

4x9

2

} + }

2

y

5

2

} = 1 12. }

x9

2

} + }

1

y

6

2

} = 1 13. }

3x0

2

} + }

y

4

2

} = 1

14. }

6x4

2

} + }

4

y

5

2

} = 1 15. 75x2 + 36y2 = 2700 16. 81x2 + 63y2 = 5103

17. GARDEN An elliptical garden is 10 feet long and 6 feet wide. Writean equation for the garden. Then graph the equation. Label the vertices,co-vertices, and foci. Assume that the major axis of the garden is horizontal.

IDENTIFYING PARTS Write the equation in standard form (if not already).

Then identify the vertices, co-vertices, and foci of the ellipse.

18. }

2x5

2

} + }

1

y

6

2

} = 1 19. }

1x2

2

1} + }

1

y

0

2

0} = 1 20. }

x4

2

} + }

y

9

2

} = 1

21. }

x9

2

} + }

2

y

5

2

} = 1 22. }

1x2

2

} + }

3

y

6

2

} = 1 23. }

2x8

2

} + }

2

y

0

2

} = 1

24. 16x2 + y2 = 16 25. 49x2 + 4y2 = 196 26. 9x2 + 100y2 = 900

27. x2 + 10y2 = 10 28. 10x2 + 25y2 = 250 29. 25x2 + 15y2 = 375


GUIDED PRACTICE


Concept Check ✓

Skill Check ✓

³ ´µµ


STUDENT HELP

Ex. 1

y

x1

1

Ex. 4

¶ · ¸ ¹ º » ¼ ½

10.4 Ellipses 613

GRAPHING Graph the equation. Then identify the vertices, co-vertices, and

foci of the ellipse.

30. }

1x6

2

} + }

3

y

6

2

} = 1 31. }

x4

2

} + }

4

y

9

2

} = 1 32. }

3x6

2

} + }

6

y

4

2

} = 1

33. }

4x9

2

} + }1

y

4

2

4} = 1 34. }

1x9

2

6} + }

1

y

0

2

0} = 1 35. }

2x5

2

6} + }

3

y

6

2

} = 1

36. }

2x2

2

5} + }

8

y

1

2

} = 1 37. }

1x2

2

1} + }

1

y

6

2

9} = 1 38. }

1x4

2

4} + }

4

y

0

2

0} = 1

39. }

4x9

2

} + }

6

y

4

2

} = 1 40. }

x4

2

} + y2 = 100 41. }

x4

2

} + }

2

y

5

2

} = }

41

}

GRAPHING In Exercises 42–50, the equations of parabolas, circles, and ellipses

are given. Graph the equation.

42. x2 + y2 = 332 43. 64x2 + 25y2 = 1600 44. 24y + x2 = 0

45. 72x2 = 144y 46. 24x2 + 24y2 = 96 47. }

8x1

2

} + }

4

9

y} = 1

48. }

31x2

2

} + }5

5

0

y

0

2

} = 1 49. }

3x6

2

} + }

3

y

6

2

} = 4 50. 5x2 + 9y2 = 45

WRITING EQUATIONS Write an equation of the ellipse with the given

characteristics and center at (0, 0).

51. Vertex: (0, 6) 52. Vertex: (0, 6) 53. Vertex: (º4, 0)Co-vertex: (5, 0) Co-vertex: (º2, 0) Co-vertex: (0, 3)

54. Vertex: (0, º7) 55. Vertex: (9, 0) 56. Vertex: (10, 0)Co-vertex: (º1, 0) Co-vertex: (0, º8) Co-vertex: (0, 4)

57. Vertex: (0, 7) 58. Vertex: (º5, 0) 59. Vertex: (0, 8)

Focus: (0, 3) Focus: (2Ï6w, 0) Focus: (0, º4Ï3w)

60. Vertex: (15, 0) 61. Vertex: (5, 0) 62. Vertex: (0, º30)Focus: (12, 0) Focus: (º3, 0) Focus: (0, 20)

63. Co-vertex: (Ï5w5w, 0) 64. Co-vertex: (0, ºÏ3w) 65. Co-vertex: (º2Ï1w0w, 0)

Focus: (0, º3) Focus: (º1, 0) Focus: (0, 9)

66. Co-vertex: (0, º3Ï3w) 67. Co-vertex: (5Ï1w1w, 0) 68. Co-vertex: (0, ºÏ7w7w)

Focus: (3, 0) Focus: (0, º7) Focus: (º2, 0)

WHISPERING GALLERY In Exercises 69–71, use the following information.

Statuary Hall is an elliptical room in the United States Capitol in Washington, D.C. The room is also called the Whispering Gallery because a person standing at one focus of the room can hear even a whisper spoken by a person standing at the other focus. This occurs because any sound that is emitted from one focus of an ellipse will reflect off the side of the ellipse to the other focus. Statuary Hall is 46 feet wide and 97 feet long.

69. Find an equation that models the shape of the room.

70. How far apart are the two foci?

71. What is the area of the floor of the room?

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 18–50Example 2: Exs. 51–68Examples 3, 4: Exs. 69–75

¶ · ¸ ¹ º » ¼ ½


72. SPACE EXPLORATION The first artificial satellite to orbit Earth wasSputnik I, launched by the Soviet Union in 1957. The orbit was an ellipse with Earth’s center as one focus. The orbit’s highest point above Earth’s surfacewas 583 miles, and its lowest point was 132 miles. Find an equation of the orbit. (Use 4000 miles as the radius of Earth.) Graph your equation.

AUSTRALIAN FOOTBALL In Exercises 73–75, use the information below.

Australian football is played on an elliptical field. The official rules state that thefield must be between 135 and 185 meters long and between 110 and 155 meterswide. c Source: The Australian News Network

73. Write an equation for the largest allowable playing field.

74. Write an equation for the smallest allowable playing field.

75. Write an inequality that describes the possible areas of an Australian football field.

76. MULTI-STEP PROBLEM A tour boat travels between two islands that are 12 milesapart. For a trip between the islands, there is enough fuel for a 20-mile tour.

a. Writing The region in which the boat can travel is bounded by an ellipse.Explain why this is so.

b. Let (0, 0) represent the center of the ellipse. Find the coordinates of each island.

c. Suppose the boat travels from one island, straightpast the other island to the vertex of the ellipse,and back to the second island. How many milesdoes the boat travel? Use your answer to find thecoordinates of the vertex.

d. Use your answers to parts (b) and (c) to write an equation for the ellipse thatbounds the region the boat can travel in.

77. LOGICAL REASONING Show that c2 = a2 º b2 for any ellipse given by the

equation }

a

x2

2} + }

b

y2

2} = 1 with foci at (c, 0) and (ºc, 0).

RATIONAL EXPONENTS Evaluate the expression without using a calculator.

(Review 7.1)

78. 1252/3 79. º85/3 80. 45/2 81. 27º2/6

82. 47/2 83. 813/4 84. 64º2/3 85. 324/5

INVERSE VARIATION The variables x and y vary inversely. Use the given values

to write an equation relating x and y. (Review 9.1)

86. x = 3, y = º2 87. x = 4, y = 6 88. x = 5, y = 1

89. x = 8, y = 9 90. x = 9, y = 2 91. x = 0.5, y = 24

GRAPHING Graph the function. State the domain and range. (Review 9.2 for 10.5)

92. ƒ(x) = }

9x

} 93. ƒ(x) = º}

9x

} 94. ƒ(x) = }

1x2}

95. ƒ(x) = }

2x4} 96. ƒ(x) = }

x1º0

2} 97. ƒ(x) = }

x +4

3}

MIXED REVIEW

Island 1 Island 2

★★Challenge

TestPreparation

¶ · ¸ ¹ º » ¼ ½

10.5 Hyperbolas 615

Hyperbolas

GRAPHING AND WRITING EQUATIONS OF HYPERBOLAS

The definition of a hyperbola is similar to that of an ellipse. For an ellipse, recall thatthe sum of the distances between a point on the ellipse and the two foci is constant.For a hyperbola, the difference is constant.

A is the set of all points Psuch that the difference of the distancesfrom P to two fixed points, called the

is constant. The line through the fociintersects the hyperbola at two points, the

The line segment joining thevertices is the and itsmidpoint is the of the hyperbola. A hyperbola has two branches and twoasymptotes. The asymptotes contain the diagonals of a rectangle centered at the hyperbola’s center, as shown below.

centertransverse axis,

vertices.

foci,

hyperbola

GOAL 1

Graph and writeequations of hyperbolas.

Use hyperbolas to solve real-life problems,such as modeling a sundial in Exs. 64–66.

. To model real-life objects,such as a sculpture in Example 4.


GOAL 2

GOAL 1


10.5

focus focus

d2d1

d2 2 d1 5 constant

P

The with center at (0, 0) is as

follows.

EQUATION TRANSVERSE AXIS ASYMPTOTES VERTICES

º = 1 Horizontal y = ±}

b

a}x (±a, 0)

º = 1 Vertical y = ±}

b

a}x (0, ±a)

The foci of the hyperbola lie on the transverse axis, c units from the center

where c2 = a2 + b2.

x2

}

b2

y2

}

a2

y2

}

b2

x2

}

a2

standard form of the equation of a hyperbola

CHARACTERISTICS OF A HYPERBOLA (CENTER AT ORIGIN)

REA

L LIFE

REA

L LIFE¾(2b, 0) (b, 0)

focus:(0, 2c)

focus:(0, c)

vertex:(0, a)

vertex:(0, 2a)

transverseaxis

¿

Hyperbola with vertical transverse axis

º = 1x 2

}

b 2

y 2

}

a 2

Hyperbola with horizontal transverse axis

º = 1y 2

}

b 2

x 2

}

a 2

¿À(0, b)

(0, 2b)

focus:(2c, 0)

focus:(c, 0)

vertex:(2a, 0)

vertex:(a, 0)

transverseaxis

Á Â Ã Ä Å Æ Ç È


Graphing an Equation of a Hyperbola

Draw the hyperbola given by 4x2 º 9y2 = 36.

SOLUTION

First rewrite the equation in standard form.

4x2 º 9y2 = 36 Write original equation.

}

43x6

2} º }

93y

6

2

} = }

33

66} Divide each side by 36.

}

x9

2} º }

y

4

2

} = 1 Simplify.

Note from the equation that a2 = 9 and b2 = 4, so a = 3 and b = 2. Because the x2-term is positive, the transverse axis is horizontal and the vertices are at (º3, 0)and (3, 0). To draw the hyperbola, first draw a rectangle that is centered at the origin, 2a = 6 units wide and 2b = 4 units high. Then show the asymptotes by drawing thelines that pass through opposite corners of the rectangle. Finally, draw the hyperbola.

Writing an Equation of a Hyperbola

Write an equation of the hyperbola with foci at (0, º3) and (0, 3) and vertices at (0, º2) and (0, 2).

SOLUTION

The transverse axis is vertical because the foci andvertices lie on the y-axis. The center is the originbecause the foci and the vertices are equidistant fromthe origin. Since the foci are each 3 units from thecenter, c = 3. Similarly, because the vertices are each 2 units from the center, a = 2.

You can use these values of a and c to find b.

b2 = c2 º a2

b2 = 32 º 22 = 9 º 4 = 5

b = Ï5w

Because the transverse axis is vertical, the standard form of the equation is as follows.

}

2

y2

2}º = 1 Substitute 2 for a and Ï5w for b.

}

y

4

2

} º }

x5

2} = 1 Simplify.

x2

}

E X A M P L E 2

E X A M P L E 1

¿ ÀÉ(0, 2)

(0, 22)

(23, 0) (3, 0)

¿ ÀÉ(0, 2)

(0, 22)

(23, 0) (3, 0)

¿ÀÊË

sÏ5, 0 d

(0, 23)

(0, 3)(0, 2)

(0, 22)

s2Ï5, 0d

Á Â Ã Ä Å Æ Ç È

10.5 Hyperbolas 617

USING HYPERBOLAS IN REAL LIFE

Using a Real-Life Hyperbola

PHOTOGRAPHY A hyperbolic mirror can be used to take panoramic photographs. A camera is pointed toward the vertex of the mirror and is positioned so that the lens is at one focus of the mirror. An equation for the cross section of the mirror

is }1y

6

2

} º }

x9

2} = 1 where x and y are measured in inches. How far from the mirror is

the lens?

SOLUTION

Notice from the equation that a2 = 16 and b2 = 9, so a = 4 and b = 3. Use thesevalues and the equation c2 = a2 + b2 to find the value of c.

c2 = a2 + b2Equation relating a, b, and c

c2 = 16 + 9 = 25 Substitute for a and b and simplify.

c = 5 Solve for c.

Since a = 4 and c = 5, the vertices are at (0, º4) and (0, 4) and the foci are at(0, º5) and (0, 5). The camera is below the mirror, so the lens is at (0, º5) and thevertex of the mirror is at (0, 4). The distance between these points is 4 º (º5) = 9.

c The lens is 9 inches from the mirror.

Modeling with a Hyperbola

The diagram at the right shows the hyperbolic cross section of a sculpture located at the Fermi National Accelerator Laboratory in Batavia, Illinois.

a. Write an equation that models the curved sides of the sculpture.

b. At a height of 5 feet, how wide is the sculpture? (Each unitin the coordinate plane represents 1 foot.)

SOLUTION

a. From the diagram you can see that the transverse axis is horizontal and a = 1. So the equation has this form:

}

1

x2

2} º }

b

y2

2} = 1

Because the hyperbola passes through the point (2, 13), you can substitute x = 2 andy = 13 into the equation and solve for b. When you do this, you obtain b ≈ 7.5.

c An equation of the hyperbola is }

1

x2

2} º }(7

y

.5

2

)2} = 1.

b. At a height of 5 feet above the ground, y = º8. To find the width of the sculpture,substitute this value into the equation and solve for x. You get x ≈ 1.46.

c At a height of 5 feet, the width is 2x ≈ 2.92 feet.

E X A M P L E 4

E X A M P L E 3

GOAL 2

¿ÀÊ(21, 0) (1, 0)

(22, 13) (2, 13)

(22, 213) (2, 213)

ÌRE

AL LIFE

RE

AL LIFESculpture

PANORAMIC

CAMERAS

A panoramic photographtaken with the camerashown above gives a 360˚view of a scene.

APPLICATION LINK


INT

ERNET

RE

AL LIFE

RE

AL LIFE

FOCUS ON

APPLICATIONS

Í Î Ï Ð Ñ Ò Ó Ô


1. Complete these statements: The points (0, º2) and(0, 2) in the graph at the right are the

ooo

? of thehyperbola. The segment joining these two points is the

ooo

? .

2. How are the definitions of ellipse and hyperbolaalike? How are they different?

3. How do the asymptotes of a hyperbola help youdraw the hyperbola?

Graph the equation. Identify the foci and asymptotes.

4. }4x9

2} º }8

y

1

2

} = 1 5. }1y

0

2

0} º }7x5

2} = 1 6. }6

x4

2} º y2 = 1

7. 36x2 º 4y2 = 144 8. 12y2 º 25x2 = 300 9. y2 º 9x2 = 9

Write an equation of the hyperbola with the given foci and vertices.

10. Foci: (0, º5), (0, 5) 11. Foci: (º8, 0), (8, 0)Vertices: (0, º3), (0, 3) Vertices: (º7, 0), (7, 0)

12. Foci: (ºÏ3w4w, 0), (Ï3w4w, 0) 13. Foci: (0, º9), (0, 9)Vertices: (º5, 0), (5, 0) Vertices: (0, º3Ï5w), (0, 3Ï5w)

14. PHOTOGRAPHY Look back at Example 3. Suppose a mirror has a cross

section modeled by the equation }2x5

2} º }

y

9

2

} = 1 where x and y are measured in

inches. If you place a camera with its lens at the focus, how far is the lens from the vertex of the mirror?


15. }1x6

2} º }

y

4

2

} = 1 16. }

y

4

2

} º }

x2

2} = 1 17. }1

y

6

2

} º }

x4

2} = 1 18. }

x4

2} º }

y

2

2

} = 1

A. B.

C. D. ÕÖÖ ×ÕÖÖ × ÕØØ ×ÕØØ ×PRACTICE AND APPLICATIONS

GUIDED PRACTICE


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Skill Check ✓

× ÕØØ (0, 2)

(0, 22)


STUDENT HELP

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 15–55Example 2: Exs. 56–63Example 3: Ex. 67Example 4: Exs. 64–66

Ex. 1

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10.5 Hyperbolas 619

STANDARD FORM Write the equation of the hyperbola in standard form.

19. 36x2 º 9y2 = 324 20. y2 º 81x2 = 81 21. 36y2 º 4x2 = 9

22. 16y2 º 36x2 + 9 = 0 23. y2 º }3x6

2} = 4 24. }

x9

2} º }

49y2

} = 9

IDENTIFYING PARTS Identify the vertices and foci of the hyperbola.

25. }

x9

2} º }6

y

4

2

} = 1 26. }4y

9

2

} º x2 = 1 27. }1x2

2

1} º }

y

4

2

} = 1

28. 4y2 º 81x2 = 324 29. 25y2 º 4x2 = 100 30. 36x2 º 10y2 = 360

GRAPHING Graph the equation. Identify the foci and asymptotes.

31. }2x5

2} º }1

y

2

2

1} = 1 32. }3x6

2} º y2 = 1 33. }2

y

5

2

} º }4x9

2} = 1

34. }

y

9

2

} º }1x0

2

0} = 1 35. }1x6

2

9} º }1y

6

2

} = 1 36. }6y

4

2

} º x2 = 1

37. }

1265x2} º }8

y

1

2

} = 1 38. }1x4

2

4} º }1y

2

2

1} = 1 39. }6x4

2} º }

94y2

} = 1

40. }2y

5

2

} º }1x6

2} = 16 41. 100x2 º 81y2 = 8100 42. x2 º 9y2 = 25

GRAPHING HYPERBOLAS Use a graphing calculator to graph the equation.

Tell what two equations you entered into the calculator.

43. }1y

4

2

4} º }1x0

2

0} = 1 44. }1x6

2} º }2

y

5

2

} = 1 45. }42x.

2

25} º }72y

.

2

25} = 1

46. }2y

.7

2

3} º }3x.5

2

8} = 1 47. }1x0

2

.1} º }2y

2

2

.3} = 1 48. 1.2x2 º 8.5y2 = 4.6

49. CRITICAL THINKING Suppose you tried to graph an equation of a hyperbola ona graphing calculator. You enter one function correctly, but you forget to enter theother function. Sketch what your graph might look like if the transverse axis ishorizontal. Then sketch what your graph might look like if the transverse axis is vertical.

GRAPHING CONIC SECTIONS In Exercises 50–55, the equations of parabolas,

circles, ellipses, and hyperbolas are given. Graph the equation.

50. }1x6

2

9} º }2y

5

2

} = 1 51. x2 + y2 = 30 52. }

y

9

2

} º }6x4

2} = 1

53. x2 = 15y 54. }1x9

2

6} + }2y

5

2

6} = 1 55. 14x2 + 14y2 = 126

WRITING EQUATIONS Write an equation of the hyperbola with the given foci

and vertices.


58. Foci: (º4 , 0), (4, 0) 59. Foci: (º6, 0), (6, 0)Vertices: (º1, 0), (1, 0) Vertices: (º5, 0), (5, 0)

60. Foci: (0, º7), (0, 7) 61. Foci: (0, º9), (0, 9)Vertices: (0, º3), (0, 3) Vertices: (0, º8), (0, 8)

62. Foci: (º8 , 0), (8, 0) 63. Foci: (0, º5Ï6w), (0, 5Ï6w)Vertices: (º4Ï3w, 0), (4Ï3w, 0) Vertices: (0, º4), (0, 4)

HOMEWORK HELP

Visit our Web sitewww.mcdougallittell.comfor help with problemsolving in Ex. 49.

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STUDENT HELP

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SUNDIAL In Exercises 64–66, use the following information.

The sundial at the left was designed by Professor John Shepherd. The shadow of thegnomon traces a hyperbola throughout the day. Aluminum rods form the hyperbolastraced on the summer solstice, June 21, and the winter solstice, December 21.

64. One focus of the summer solstice hyperbola is 207 inches above the ground. Thevertex of the aluminum branch is 266 inches above the ground. If the x-axis is355 inches above the ground and the center of the hyperbola is at the origin,write an equation for the summer solstice hyperbola.

65. One focus of the winter solstice hyperbola is 419 inches above the ground. Thevertex of the aluminum branch is 387 inches above the ground and the center ofthe hyperbola is at the origin. If the x-axis is 355 inches above the ground, writean equation for the winter solstice hyperbola.

66. Use your equations from Exercises 64 and 65 to draw the lower branch of thesummer solstice hyperbola and the upper branch of the winter solstice hyperbola.

67. AERONAUTICS When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane. If the airplane is flying parallel to the ground, the sound waves intersect the ground in a hyperbola with the airplane directly above its center. A sonic boom is heard along the hyperbola. If you hear a sonic boom that is audible along a hyperbola

with the equation }1x0

2

0} º }

y

4

2

} = 1 where x and y are

measured in miles, what is the shortest horizontal distance you could be to the airplane?

68. MULTI-STEP PROBLEM Suppose you are making a ring out of clay for a necklace. If you have a fixed volume of clay and you want the ring to have a certain thickness, the area of the ring becomes fixed. However, you can still vary the inner radius x and the outer radius y.

a. Suppose you want to make a ring with an area of 2 square inches. Write an equation relating x and y.

b. Find three coordinate pairs (x, y) that satisfy the relationship from part (a). Then find the width of the ring, y º x, for each coordinate pair.

c. Writing How does the width of the ring, y º x, change as x and y both increase? Explain why this makes sense.

69. Use the diagram at the right to show that |d2 º d1| = 2a.

70. LOCATING AN EXPLOSION Two microphones, 1 mile apart, record anexplosion. Microphone A receives the sound 2 seconds after Microphone B. Is this enough information to decide where the sound came from? Use the factthat sound travels at 1100 feet per second.

SUNDIAL TheRichard D. Swensen

sundial, located at theUniversity of Wisconsin –River Falls, gives the correcttime to the minute.

APPLICATION LINK


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FOCUS ON

APPLICATIONS

★★Challenge

TestPreparation

ground

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y

x

fixed

Ex. 69

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10.5 Hyperbolas 621

GRAPHING FUNCTIONS Graph the function. (Review 2.8, 5.1 for 10.6)

71. y = 2|x + 4| + 1 72. y = |x º 4| + 5 73. y = º|x º 6| º 8

74. y = 3(x º 1)2 + 7 75. y = º2(x º 3)2 º 6 76. y = }

12}(x + 4)2 + 5

WRITING FUNCTIONS Write a polynomial function of least degree that has real

coefficients, the given zeros, and a leading coefficient of 1. (Review 6.7)

77. 3, 1, 2 78. º7, º1, 3 79. 6, º2, 2

80. º6, 4, 2 81. 5, i, ºi 82. 3, º3, 2i

EVALUATING LOGARITHMIC EXPRESSIONS Evaluate the expression without

using a calculator. (Review 8.4)

83. log 10,000 84. log3 27 85. log5 625 86. log2 128

87. log4 64 88. log3 243 89. log6 216 90. log100 100,000,000

91. TEST SCORES Find the mean, median, mode(s), and range of the followingset of test scores. (Review 7.7)

63, 67, 72, 75, 77, 78, 81, 81, 85, 86, 89, 89, 91, 92, 99

Write an equation of the ellipse with the given characteristics and center at

(0, 0). (Lesson 10.4)

1. Vertex: (0, 7) 2. Vertex: (º6, 0) 3. Vertex: (º10, 0)Co-vertex: (º3, 0) Co-vertex: (0, º1) Focus: (6, 0)

4. Vertex: (0, 5) 5. Co-vertex: (0, 2Ï3w) 6. Co-vertex: (º9, 0)Focus: (0, Ï1w7w) Focus: (ºÏ3w, 0) Focus: (0, 4)

Graph the equation. Identify the vertices, co-vertices, and foci. (Lesson 10.4)

7. }

x4

2} + }4

y

9

2

} = 1 8. }

x6

2} + y2 = 1 9. x2 + 9y2 = 36

Write an equation of the hyperbola with the given characteristics. (Lesson 10.5)


12. Foci: (º6, 0), (6, 0) 13. Foci: (0, º2Ï5w), (0, 2Ï5w)Vertices: (º4, 0), (4, 0) Vertices: (0, º4), (0, 4)

Graph the equation. Identify the vertices, foci, and asymptotes. (Lesson 10.5)

14. }2y

5

2

} º }3x6

2} = 1 15. 8y2 º 20x2 = 160 16. 18x2 º 4y2 = 36

17. SPACE EXPLORATION Suppose a satellite’s orbit is an ellipse with Earth’scenter at one focus. If the satellite’s least distance from Earth’s surface is150 miles and its greatest distance from Earth’s surface is 600 miles, write anequation for the ellipse. (Use 4000 miles as Earth’s radius.) (Lesson 10.4)

QUIZ 2 Self-Test for Lessons 10.4 and 10.5

MIXED REVIEW

ã ä å æ ç è é êDeveloping Concepts

ACTIVITY 10.6Group Activity for use with Lesson 10.6

GROUP ACTIVITY

Work with a partner.

MATERIALS

• flashlight

• graph paper

• pencil

Exploring Conic Sections

c QUESTION How do a plane and a double-napped cone intersect to form

different conic sections?

c EXPLORING THE CONCEPT The reason that parabolas, circles, ellipses,

and hyperbolas are called conics or conic sections is that each can be formed

by the intersection of a plane and a double-napped cone, as shown below.

The beam of light from a flashlight is a cone. When the light hits a flat surface such

as a wall, the edge of the beam of light forms a conic section.

Work in a group to find an equation of a conic formed by a flashlight beam.

On a piece of graph paper, draw x- and y-axes to make a coordinate plane.

Tape the paper to a wall.

Aim a flashlight perpendicular to the

paper so that the light forms a circle.

Move the flashlight so that the circle is

centered on the origin of the coordinate

plane.

Holding the flashlight very still, trace

the circle on the graph paper. Find the

radius of the circle and use it to write

the standard form of the equation of

the circle.

Aim the flashlight at the paper to form

an ellipse with a vertical major axis and

center at the origin. Trace the ellipse and

find the standard form of its equation.

c DRAWING CONCLUSIONS

1. Compare the equation of your circle with the equations found by other groups.

Are your equations all the same? Why or why not?

2. Compare the equation of your ellipse with the equations found by other groups.

Are your equations all the same? Why or why not?

5

4

3

2

1

Circle Ellipse Parabola Hyperbola


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(23, 1)

(22, 1)

Graphing and ClassifyingConics

WRITING AND GRAPHING EQUATIONS OF CONICS

Parabolas, circles, ellipses, and hyperbolas are all curves that are formed by theintersection of a plane and a double-napped cone. Therefore, these shapes are called

or simply

In previous lessons you studied equations of parabolas with vertices at the origin and equations of circles, ellipses, and hyperbolas with centers at the origin. In this lesson you will study equations of conics that have been translated in thecoordinate plane.

Writing an Equation of a Translated Parabola

Write an equation of the parabola whose vertex is at (º2, 1) and whose focus is at (º3, 1).

SOLUTION

Choose form: Begin by sketching the parabola, as shown. Because the parabola opens to the left, it has the form

(y º k)2 = 4p(x º h)

where p < 0.

Find h and k: The vertex is at (º2, 1), so h = º2 and k = 1.

Find p: The distance between the vertex (º2, 1) and the focus (º3, 1) is

|p| = Ï(ºw3w ºw (wºw2w))w2w+w (w1w ºw 1w)2w= 1

so p = 1 or p = º1. Since p < 0, p = º1.

c The standard form of the equation is (y º 1)2 = º4(x + 2).

E X A M P L E 1

conics.conic sections

GOAL 1

10.6 Graphing and Classifying Conics 623

Write and graph an

equation of a parabola with

its vertex at (h, k) and an

equation of a circle, ellipse,

or hyperbola with its center

at (h, k).

Classify a conic

using its equation, as applied

in Example 8.

. To model real-lifesituations involving more

than one conic, such as the

circles that an ice skater

uses to practice figure eights

in Ex. 64.


GOAL 2

GOAL 1


10.6R

EAL LIFE

REA

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In the following equations the point (h, k) is the vertex of the parabola and

the center of the other conics.

CIRCLE (x º h)2 + (y º k)2 = r2

Horizontal axis Vertical axis

PARABOLA (y º k)2 = 4p(x º h) (x º h)2 = 4p(y º k)

ELLIPSE }(x º

a2

h)2

} + = 1 + = 1

HYPERBOLA }(x º

a2

h)2

} º = 1 º }(x º

b2

h)2

} = 1(y º k)2

}

a2

(y º k)2

}

b2

(y º k)2

}

a2

(x º h)2

}

b2

(y º k)2

}

b2

STANDARD FORM OF EQUATIONS OF TRANSLATED CONICS

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Graphing the Equation of a Translated Circle

Graph (x º 3)2 + (y + 2)2 = 16.

SOLUTION

Compare the given equation to the standard form of the equation of a circle:

(x º h)2 + (y º k)2 = r2

You can see that the graph is a circle with center at (h, k) = (3, º2) and radius r = 4.

Plot the center.

Plot several points that are each 4 units from thecenter:

(3 + 4, º2) = (7, º2)

(3 º 4, º2) = (º1, º2)

(3, º2 + 4) = (3, 2)

(3, º2 º 4) = (3, º6)

Draw a circle through the points.

Writing an Equation of a Translated Ellipse

Write an equation of the ellipse with foci at (3, 5) and (3, º1) and vertices at (3, 6)and (3, º2).

SOLUTION

Plot the given points and make a rough sketch. The ellipse has a vertical major axis, so its equation is of this form:

+ }(y º

a2

k)2

} = 1

Find the center: The center is halfway between the vertices.

(h, k) = S}3 +

23

}, }6 +

2(º2)}D = (3, 2)

Find a: The value of a is the distance between the vertex and the center.

a = Ï(3w ºw 3w)2w +w (w6w ºw 2w)2w = Ï0w +w 4w2w = 4

Find c: The value of c is the distance between the focus and the center.

c = Ï(3w ºw 3w)2w +w (w5w ºw 2w)2w = Ï0w +w 3w2w = 3

Find b: Substitute the values of a and c into the equation b2 = a2 º c2.

b2 = 42 º 32

b2 = 7

b = Ï7w

c The standard form of the equation is }(x º

73)2

} + }(y º

162)2

} = 1.

(x º h)2}

b2

E X A M P L E 3

E X A M P L E 2

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(3, 2)

(7, 22)

(3, 26)

(3, 22)

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(3, 22)

(3, 6)

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Graphing the Equation of a Translated Hyperbola

Graph (y + 1)2 º }(x +

41)2

} = 1.

SOLUTION

The y2-term is positive, so the transverse axis is vertical.Since a2 = 1 and b2 = 4, you know that a = 1 and b = 2.

Plot the center at (h, k) = (º1, º1). Plot the vertices 1 unit above and below the center at (º1, 0) and (º1, º2).

Draw a rectangle that is centered at (º1, º1) and is 2a = 2 units high and 2b = 4 units wide.

Draw the asymptotes through the corners of the rectangle.

Draw the hyperbola so that it passes through the vertices and approaches theasymptotes.

Using Circular Models

COMMUNICATIONS A cellular phone transmission tower located 10 miles west and5 miles north of your house has a range of 20 miles. A second tower, 5 miles east and10 miles south of your house, has a range of 15 miles.

a. Write an inequality that describes each tower’s range.

b. Do the two regions covered by the towers overlap?

SOLUTION

a. Let the origin represent your house. The first tower is at (º10, 5) and theboundary of its range is a circle with radius 20. Substitute º10 for h, 5 for k,and 20 for r into the standard form of the equation of a circle.

(x º h)2 + (y º k)2 = r2 Standard form of a circle

(x + 10)2 + ( y º 5)2 < 400 Region inside the circle

The second tower is at (5, º10). The boundary of its range is a circle with radius 15.

(x º h)2 + (y º k)2 = r2 Standard form of a circle

(x º 5)2 + ( y + 10)2 < 225 Region inside the circle

b. One way to tell if the regions overlap is to graph theinequalities. You can see that the regions do overlap.

You can also check whether the distance betweenthe two towers is less than the sum of the ranges.

Ï(ºw1w0w ºw 5w)2w +w (w5w ºw (wºw1w0w))w2w <? 20 + 15

15Ï2w <? 35

21.2 < 35 ✓

c The regions do overlap.

E X A M P L E 5

E X A M P L E 4

CELLULAR

PHONES work onlywhen there is a transmissiontower nearby to retrieve thesignal. Because of the needfor many towers, they areoften designed to blend inwith the environment.

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CLASSIFYING A CONIC FROM ITS EQUATION

The equation of any conic can be written in the form

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

which is called a in x and y. The expressionB2 º 4AC is called the of the equation and can be used to determinewhich type of conic the equation represents.

Classifying a Conic

a. Classify the conic given by 2x2 + y2 º 4x º 4 = 0.

b. Graph the equation in part (a).

SOLUTION

a. Since A = 2, B = 0, and C = 1, the value of the discriminant is as follows:

B2 º 4AC = 02 º 4(2)(1) = º8

c Because B2 º 4AC < 0 and A ≠ C, the graph is an ellipse.

b. To graph the ellipse, first complete the square as follows.

2x2 + y2 º 4x º 4 = 0

(2x2 º 4x) + y2 = 4

2(x2 º 2x) + y2 = 4

2(x2 º 2x + ooo

? ) + y2 = 4 + 2(ooo

? )

2(x2 º 2x + 1) + y2 = 4 + 2(1)

2(x º 1)2 + y2 = 6

}(x º

31)2

} + }y

6

2

} = 1

By comparing this equation to }(x º

b2

h)2

} + }(y º

a2

k)2

} = 1, you can see that h = 1,

k = 0, a = Ï6w, and b = Ï3w. Use these facts to draw the ellipse.

E X A M P L E 6

discriminantgeneral second-degree equation

GOAL 2

Look Back For help with completingthe square, see p. 282.

STUDENT HELP

��(1, 0) �� s1, Ï6 d

s1, 2Ï6 d

s1 2 Ï3 , 0 d s1 1 Ï3, 0 d

If the graph of Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 is a conic, then the type of

conic can be determined as follows.

DISCRIMINANT TYPE OF CONIC

B2 º 4AC < 0, B = 0, and A = C Circle

B2 º 4AC < 0 and either B ≠ 0 or A ≠ C Ellipse

B2 º 4AC = 0 Parabola

B2 º 4AC > 0 Hyperbola

If B = 0, each axis of the conic is horizontal or vertical. If B ≠ 0, the axes are

neither horizontal nor vertical.

CLASSIFYING CONICSCONCEPT

SUMMARY

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Classifying a Conic

a. Classify the conic given by 4x2 º 9y2 + 32x º 144y º 548 = 0.

b. Graph the equation in part (a).

SOLUTION

a. Since A = 4, B = 0, and C = º9, the value of the discriminant is as follows:

B2 º 4AC = 02 º 4(4)(º9) = 144

c Because B2 º 4AC > 0, the graph is a hyperbola.

b. To graph the hyperbola, first complete the square as follows.

4x2 º 9y2 + 32x º 144y º 548 = 0

(4x2 + 32x) º (9y2 + 144y) = 548

4(x2 + 8x + ooo

? ) º 9(y2 + 16y + ooo

? ) = 548 + 4(ooo

? ) º 9(ooo

? )

4(x2 + 8x + 16) º 9(y2 + 16y + 64) = 548 + 4(16) º 9(64)

4(x + 4)2 º 9(y + 8)2 = 36

}(x +

32

4)2

} º = 1

By comparing this equation to }(x –

a2

h)2

} º }(y º

b2

k)2

} = 1, you can see that h = º4,

k = º8, a = 3, and b = 2.

To draw the hyperbola, plot the center at(h, k) = (º4, º8) and the vertices at (º7, º8)and (º1, º8). Draw a rectangle 2a = 6 unitswide and 2b = 4 units high and centered at(º4, º8). Draw the asymptotes through thecorners of the rectangle. Then draw thehyperbola so that it passes through the verticesand approaches the asymptotes.

Classifying Conics in Real Life

The diagram at the right shows the mirrors in a Cassegrain telescope. The equations of the two mirrors are given below. Classify each mirror as parabolic, elliptical, or hyperbolic.

a. Mirror A: y2 º 72x º 450 = 0

b. Mirror B: 88.4x2 º 49.7y2 º 4390 = 0

SOLUTION

EQUATION B2 º 4AC TYPE OF MIRROR

a. y2 º 72x º 450 = 0 02 º 4(0)(1) = 0 Parabolic

b. 88.4x2 º 49.7y2 º 4390 = 0 02 º 4(88.4)(º49.7) > 0 Hyperbolic

E X A M P L E 8

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22

E X A M P L E 7

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mirror B

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1. Explain why circles, ellipses, parabolas, and hyperbolas are called conic sections.

2. How are the graphs of x2 + y2 = 25 and (x º 1)2 + (y + 2)2 = 25 alike?How are they different?

3. How can the discriminant B2 º 4AC be used to classify the graph ofAx2 + Bxy + Cy2 + Dx + Ey + F = 0?

Write an equation for the conic section.

4. Circle with center at (4, º1) and radius 7

5. Ellipse with foci at (2, º4) and (5, º4) and vertices at (º1, º4) and (8, º4)

6. Parabola with vertex at (3, º2) and focus at (3, º4)

7. Hyperbola with foci at (5, 2) and (5, º6) and vertices at (5, 0) and (5, º4)

Classify the conic section.

8. x2 + 2x º 4y + 4 = 0 9. 3x2 º 5y2 º 6x + y º 2 = 0

10. x2 + y2 + 7x º 4y º 8 = 0 11. º5x2 º 2y2 + x º 3y + 1 = 0

12. COMMUNICATIONS Look back at Example 5. Suppose there is a tower 25 miles east and 30 miles north of your house with a range of 25 miles. Does the region covered by this tower overlap the regions covered by the two towers in Example 5? Illustrate your answer with a graph.

WRITING EQUATIONS Write an equation for the conic section.

13. Circle with center at (9, 3) and radius 4

14. Circle with center at (º4, 2) and radius 3

15. Parabola with vertex at (1, º2) and focus at (1, 1)

16. Parabola with vertex at (º3, 1) and directrix x = º8

17. Ellipse with vertices at (2, º3) and (2, 6) and foci at (2, 0) and (2, 3)

18. Ellipse with vertices at (º2, 2) and (4, 2) and co-vertices at (1, 1) and (1, 3)

19. Hyperbola with vertices at (5, º4) and (5, 4) and foci at (5, º6) and (5, 6)

20. Hyperbola with vertices at (º4, 2) and (1, 2) and foci at (º7, 2) and (4, 2)

GRAPHING Graph the equation. Identify the important characteristics of the

graph, such as the center, vertices, and foci.

21. (x º 6)2 + (y º 2)2 = 4 22. (x + 7)2 = 12(y º 3)

23. }(y º

16

8)2

} º }(x +

43)2

} = 1 24. }(x º

253)2

} + }(y +

496)2

} = 1

25. + }y

9

2

} = 1 26. }1x6

2} º (y + 4)2 = 1

27. (x + 7)2 + (y º 1)2 = 1 28. (y º 4)2 = 3(x + 2)

(x + 1)2

}16


GUIDED PRACTICE


Skill Check ✓

STUDENT HELP

HOMEWORK HELP

Examples 1, 3: Exs. 13–20Examples 2, 4: Exs. 21–28Example 5: Exs. 63, 64Examples 6, 7: Exs. 29–62Example 8: Exs. 65–67


STUDENT HELP

" # $ % & ' ( )


CLASSIFYING Classify the conic section.

29. 9x2 + 4y2 + 36x º 24y + 36 = 0 30. x2 º 4y2 + 3x º 26y º 30 = 0

31. 4x2 º 9y2 + 18y + 3x = 0 32. x2 + y2 º 10x º 2y + 10 = 0

33. 36x2 + 16y2 º 25x + 22y + 2 = 0 34. 4x2 + 4y2 º 16x + 4y º 60 = 0

35. 9y2 º x2 + 2x + 54y + 62 = 0 36. 16x2 + 25y2 º 18x º 20y + 8 = 0

37. x2 º 2x + 8y + 9 = 0 38. 2y2 º 8y º 4x + 10 = 0

39. 12x2 + 20y2 º 12x + 40y º 37 = 0 40. 9x2 º y2 + 54x + 10y + 55 = 0

41. x2 + y2 º 4x º 2y º 4 = 0 42. 16x2 + 9y2 + 24x º 36y + 23 = 0

43. 16y2 º x2 + 2x + 64y + 63 = 0 44. x2 º 4x + 16y + 17 = 0


45. 9x2 º 4y2 + 36x º 24y º 36 = 0 46. y2 º 2y º 4x + 9 = 0

47. 9x2 + 4y2 + 36x + 24y + 36 = 0 48. y2 º x2 + 6y + 4x + 4 = 0

49. 4x2 + 9y2 º 16x + 54y + 61 = 0 50. x2 + y2 º 4x + 6y + 4 = 0

A. B. C.

D. E. F.

CLASSIFYING AND GRAPHING Classify the conic section and write its

equation in standard form. Then graph the equation.

51. y2 º 12y + 4x + 4 = 0 52. x2 + y2 º 6x º 8y + 24 = 0

53. 9x2 º y2 º 72x + 8y + 119 = 0 54. 4x2 + y2 º 48x º 4y + 48 = 0

55. x2 + 4y2 º 2x º 8y + 1 = 0 56. x2 + y2 º 12x º 24y + 36 = 0

57. 16x2 º y2 + 16y º 128 = 0 58. x2 + 9y2 + 8x + 4y + 7 = 0

59. x2 + y2 º 12x º 12y + 36 = 0 60. y2 º 2x º 20y + 94 = 0

61. x2 + 4x º 8y + 12 = 0 62. º9x2 + 4y2 º 36x º 16y º 164 = 0

63. WHISPER DISHES The whisper dish shown at the left can be seen at the Thronateeska Discovery Center in Albany, Georgia. Two dishes arepositioned so that their vertices are 50 feet apart. The focus of each dish is 3 feet from its vertex. Write equations for the cross sections of the dishes so that the vertex of one dish is at the origin and the vertex of the other dish is on the positive x-axis.

*2

+ , --2

-, **2 . ,

2+

, *+2 /

0-- , *-- , *

WHISPER DISHES

are two parabolicdishes set up facing directlytoward each other. A personlistening at the focus of onedish is able to hear even thesoftest sound made at thefocus of the other dish.

RE

AL LIFE

RE

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FOCUS ON

APPLICATIONS

" # $ % & ' ( )


64. FIGURE SKATING To practice making a figure eight, a figure skater willskate along two circles etched in the ice. Write equations for two externallytangent circles that are each 6 feet in diameter so that the center of one circle is at the origin and the center of the other circle is on the positive y-axis.

65. VISUAL THINKING A new crayon has a cone-shaped tip. When it is used for the first time, a flat spot is worn on the tip. The edge of the flat spot is a conic section, as shown. What type(s) of conic could it be?

66. VISUAL THINKING When a pencil is sharpened the tip becomes a cone. On a pencil with flat sides, the intersection of the cone with each flat side is a conic section. What type of conic is it?

67. ASTRONOMY A Gregorian telescope contains two mirrors whose crosssections can be modeled by the equations 405x2 + 729y2 º 295,245 = 0 andº120y2 º 1440x = 0. What types of mirrors are they?

68. MULTIPLE CHOICE Which of the following is an equation of the hyperbolawith vertices at (3, 5) and (3, º1) and foci at (3, 7) and (3, º3)?

¡A }(x º

253)2

} º }(y º

92)2

} = 1 ¡B }(y º

92)2

} º }(x º

253)2

} = 1

¡C }(y º

92)2

} º }(x º

73)2

} = 1 ¡D }(x º

93)2

} º }(y º

162)2

} = 1

¡E }(y º

92)2

} º }(x º

163)2

} = 1

69. MULTIPLE CHOICE What conic does 25x2 + y2 º 100x º 2y + 76 = 0 represent?

¡A Parabola ¡B Circle ¡C Ellipse

¡D Hyperbola ¡E Not enough information

70. DEGENERATE CONICS A degenerate conic occurs when the intersection of a plane with a double-napped cone is something other than a parabola, circle, ellipse,or hyperbola.

a. Imagine a plane perpendicular to the axis of a double-napped cone. As the plane passes through the cone, theintersection is a circle whose radius decreases and thenincreases. At what point is the intersection somethingother than a circle? What is the intersection?

b. Imagine a plane parallel to the axis of a double-nappedcone. As the plane passes through the cone, theintersection is a hyperbola whose vertices get closertogether and then farther apart. At what point is theintersection something other than a hyperbola? What is the intersection?

c. Imagine a plane parallel to the nappe passing through a double-napped cone. As the plane passes through thecone, the intersection is a parabola that gets narrowerand then flips and gets wider. At what point is theintersection something other than a parabola? What is the intersection?

TestPreparation

★★Challenge

EXTRA CHALLENGE


nappe

conic section

conic section

" # $ % & ' ( )


SYSTEMS Solve the system using any algebraic method. (Review 3.2 for 10.7)

71. x º y = 10 72. 4x + 3y = 1 73. 4x + y = 23x º 2y = 25 º3x º 6y = 3 6x + 3y = 0

74. 2x º 3y = 0 75. 23x = 68 76. x = yx + 6y = 14 x + 3y = 19 123x º 18y = 17

EVALUATING LOGARITHMIC EXPRESSIONS Evaluate the expression. (Review 8.4)

77. log7 75 78. log4 64 79. log5 1

80. log1/3 9 81. log25 625 82. log 0.0001

SOLVING EQUATIONS Solve the equation. (Review 8.8)

83. = 20 84. = 1 85. = 7

86. }1 +

135eº6x} = 3 87. }

1 +254eº4x} = 9 88. }

1 +92eº3x} = 7

8}}

1 + 8eºx

10}}

1 + 9eº2x

40}}

1 + 6eº4x

MIXED REVIEW

History of Conic Sections

THENTHEN

NOWNOW

IN 200 B.C. conic sections were studied thoroughly for the first time bya Greek mathematician named Apollonius. Six hundred years later, theEgyptian mathematician Hypatia simplified the works of Apollonius,making it accessible to many more people. For centuries, conics werestudied and appreciated only for their mathematical beauty rather than for their occurrence in nature or practical use.

TODAY astronomers know that the paths of celestial objects, such as planetsand comets, are conic sections. For example, a comet’s path can be parabolic, hyperbolic, or elliptical.

Tell what type of path each comet follows. Which comet(s) will pass by the sun

more than once?

1. 3550x2 + 14,200x + 7100y º 13,050 = 0

2. 2200x2 + 4600y2 º 13,200x º 18,400y + 12,900 = 0

3. 5000x2 º 6500y2 + 20,000x º 52,000y º 695,000 = 0

Debra Fischer

discovers two planets.

Hypatia simplifies

Apollonius’ Conics.

Johannes Kepler discovers that

the planets’ orbits are elliptical.

APPLICATION LINK


INT

ERNET

200 B .C .

Apollonius studies

conic sections.

1999

A .D . 400

1609

1 2 3 4 5 6 7 8


Solving Quadratic Systems

SOLVING A SYSTEM OF EQUATIONS

In Lesson 3.2 you studied two algebraic techniques for solving a system of linear equations. You can use the same techniques (substitution and linearcombination) to solve quadratic systems.

Finding Points of Intersection

Find the points of intersection of the graphs of x2 + y2 = 13 and y = x + 1.

SOLUTION

To find the points of intersection, substitute x + 1 for y in the equation of the circle.

x2 + y2 = 13 Equation of circle

x2 + (x + 1)2 = 13 Substitute x + 1 for y.

x2 + x2 + 2x + 1 = 13 Expand the power.

2x2 + 2x º 12 = 0 Combine like terms.

2(x º 2)(x + 3) = 0 Factor.

x = 2 or x = º3 Zero product property

You now know the x-coordinates of the points of intersection. To find the y-coordinates, substitute x = 2 and x = º3 into the linear equation and solve for y.

c The points of intersection are (2, 3) and (º3, º2).

✓CHECK You can check your answer algebraically

by substituting the coordinates of the points into

each equation. Another way to check your answer

is to graph the two equations. You can see from the

graph shown that the line and the circle intersect in

two points, at (2, 3) and at (º3, º2).

E X A M P L E 1

GOAL 1

Solve systems of

quadratic equations.

Use quadratic

systems to solve real-lifeproblems, such as determining

when one car will catch up to

another in Ex. 58.

. To model real-lifesituations with quadratic

systems, such as finding the

epicenter of an earthquake

in Example 4.


GOAL 2

GOAL 1


10.7R

EAL LIFE

REA

L LIFE

Investigating Points of Intersection

The circle and line in Example 1 intersect in two points. A circle and a line can also intersect in one point or no points. Sketch examples to illustrate thedifferent numbers of points of intersection that the following graphs can have.

a. Circle and parabola b. Ellipse and hyperbola

c. Circle and ellipse d. Hyperbola and line

DevelopingConcepts

ACTIVITY

9: ; <(23, 22)

(2, 3)

Look Back For help with solvingsystems, see p. 148.

STUDENT HELP

= > ? @ A B C D

10.7 Solving Quadratic Systems 633

Solving a System by Substitution

Find the points of intersection of the graphs in the system.

x2 + 4y2 º 4 = 0 Equation 1

º2y2 + x + 2 = 0 Equation 2

SOLUTION

Because Equation 2 has no x2-term, solve that equation for x.

º2y2 + x + 2 = 0

x = 2y2 º 2

Next, substitute 2y2 º 2 for x in Equation 1 and solve for y.

x2 + 4y2 º 4 = 0 Equation 1

(2y2 º 2)2 + 4y2 º 4 = 0 Substitute for x.

4y4 º 8y2 + 4 + 4y2 º 4 = 0 Expand the power.

4y4 º 4y2 = 0 Combine like terms.

4y2(y2 º 1) = 0 Factor common

monomial.

4y2(y º 1)(y + 1) = 0 Difference of squares.

y = 0, y = 1, or y = º1 Zero product property

The corresponding x-values are x = º2, x = 0, and x = 0.

c The graphs intersect at (º2, 0), (0, 1), and (0, º1), as shown.

Solving a System by Linear Combination

Find the points of intersection of the graphs in the system.

x2 + y2 º 16x + 39 = 0 Equation 1

x2 º y2 º 9 = 0 Equation 2

SOLUTION

You can eliminate the y2-term by adding the two equations. The resulting equationcan be solved for x because it contains no other variables.

x2 + y2 º 16x + 39 = 0

x2 º y2 º 9 = 0

2x2 º 16x + 30 = 0 Add.

2(x º 3)(x º 5) = 0 Factor.

x = 3 or x = 5 Zero product

property

The corresponding y-values are y = 0 and y = ±4.

c The graphs intersect at (3, 0), (5, 4), and (5, º4), as shown.

E X A M P L E 3

E X A M P L E 2

9; <(22, 0)(0, 1)

(0, 21)

:

E: ; <(5, 24)

(3, 0)

(5, 4)

Look Back For help with factoring,see p. 256.

STUDENT HELP

= > ? @ A B C D


SEISMOLOGIST

A seismologistdetermines the location andintensity of an earthquakeusing an instrument whichmeasures energy wavesresulting from movements in the Earth’s crust.

CAREER LINK


INT

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FOCUS ON

CAREERS SOLVING QUADRATIC SYSTEMS IN REAL LIFE

Solving a System of Quadratic Models

SEISMOLOGY A seismograph measures the intensity of an earthquake. Although a seismograph can determine the distance to the earthquake’s epicenter, it cannotdetermine in what direction the epicenter is located. Use the following informationfrom three seismographs to find an earthquake’s epicenter.

Location 1: 500 miles from the epicenter

Location 2: 100 miles west and 400 miles south of Location 1400 miles from the epicenter

Location 3: 300 miles east and 600 miles south of Location 1200 miles from the epicenter

SOLUTION

Let each unit represent 100 miles. If Location 1 is at (0, 0), then Location 2 is at (º1, º4) and Location 3 is at (3, º6). Write the equation of each circle.

Location 1: x2 + y2 = 25

Location 2: (x + 1)2 + (y + 4)2 = 16, orx2 + 2x + 1 + y2 + 8y + 16 = 16

Location 3: (x º 3)2 + (y + 6)2 = 4, orx2 º 6x + 9 + y2 + 12y + 36 = 4

Subtract the equation for Location 1 from the equation for Location 2.

x2 + 2x + 1 + y2 + 8y + 16 = 16

º (x2 + y2 = 25)

2x + 8y + 17 = º9

2x + 8y = º26, or x + 4y = º13

Then subtract the equation for Location 1 from the equation for Location 3.

x2 º 6x + 9 + y2 + 12y + 36 = 4

º (x2 + y2 = 25)

º6x + 12y + 45 = º21

º6x + 12y = º66, or ºx + 2y = º11

You are left with two linear equations. Solve this linear system to find the epicenter.

x + 4y = º13ºx + 2y = º11

6y = º24

y = º4

x = 3

c The epicenter of the earthquake is 300 miles east and 400 miles south of Location 1.

E X A M P L E 4

GOAL 2

:: ;<1

2

3

= > ? @ A B C D


1. Complete this statement: The equations x2 + 3y2 º 2y = 4 and x2 + y2 = 5 are

an example of a(n) ooo

? system.

2. Sketch an example of a circle and a line intersecting in a single point.

3. Explain what method you would use to find the points of intersection of thegraphs in the following system. Do not solve the system.

4x2 + y2 º 16x = 0 Equation 1

x2 º y2 + 7 = 0 Equation 2

Find the points of intersection, if any, of the graphs in the system.

4. x2 + y2 = 17 5. x2 + y2 + 8x º 20y + 7 = 0

y = x + 3 x2 + 9y2 + 8x + 4y + 7 = 0

6. x2 + y2 º 3x = 8 7. x2 º 2x + 2y + 2 = 0

2x2 º y2 = 10 ºx2 + 2x º y + 3 = 0

8. SEISMOLOGY Look back at Example 4. Why are three (not just two)seismographs needed to determine the location of the epicenter?

CHECKING POINTS OF INTERSECTION Determine whether the given point is a

point of intersection of the graphs in the system.

9. x2 + y2 = 25 10. x2 + y2 = 41 11. x2 + 4x º 4y º 16 = 0

y = º3 y = ºx º 1 º2x + y + 1 = 0

Point: (º3, 4) Point: (4, º5) Point: (6, 11)

12. 3x2 º 5y2 + 2y = 45 13. 2x2 º 4y = 22 14. 6x2 º 5x + 8y2 + y = 23

y = 2x + 10 y = º2x + 3 y = x º 1

Point: (º3, 4) Point: (º5, 7) Point: (2, 1)

SOLVING SYSTEMS Find the points of intersection, if any, of the graphs in

the system.

15. x2 º y = 5 16. x2 + y2 = 18 17. º3x2 + y2 = 9º3x + y = º7 x º y = 0 º2x + y = 0

18. 9x2 + 4y2 = 36 19. x2 + y2 = 5 20. x + 2y2 = º6ºx + y = º4 y = º2x x + 8y = 0

21. 5x2 + 3y2 = 17 22. 4x2 º 5y2 = 16 23. 2x2 + 2y2 = 15ºx + y = º1 3x + y = 6 x + 2y = 6

24. x2 + y2 = 1 25. x2 + y2 = 20 26. x2 + y2 = 5x + y = º1 y = x º 4 y = 3x + 5

27. x2 = 6y 28. x2 + y2 = 9 29. x2 + y2 = 7y = ºx x º 3y = 3 y = x º 7

30. y2 º 2x2 = 6 31. 6x2 + 3y2 = 12 32. 3x2 º y2 = º6y = º2x y = ºx + 2 y = 2x + 1


GUIDED PRACTICE


Concept Check ✓

Skill Check ✓

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 9–32Examples 2, 3: Exs. 33–51Example 4: Exs. 52–55,

58–63


STUDENT HELP

F G H I J K L M


SOLVING SYSTEMS Find the points of intersection, if any, of the graphs in the system.

33. x2 + y2 = 16 34. º3x2 + y2 º 3x = 0 35. ºx2 + y2 + 10 = 0x2 º 5y = 5 x2 º y2 + 27 = 0 º3y2 + x + 1 = 0

36. x2 + 2y2 º 10 = 0 37. y2 = 16x 38. 10y = x2

4y2 + x + 4 = 0 4x º y = º24 x2 º 6 = º2y

39. y2 + x = 2 40. x2 º 16y2 = 16 41. x2 + y2 = 813x + y = 8 x2 + y2 = 9 x + y = 0

42. 16x2 º y2 + 16y º 128 = 0 43. x2 º y2 º 8x + 8y º 24 = 0y2 º 48x º 16y º 32 = 0 x2 + y2 º 8x º 8y + 24 = 0

44. x2 + 4y2 º 4x º 8y + 4 = 0 45. 4x2 º 56x + 9y2 + 160 = 0x2 + 4y º 4 = 0 4x2 + y2 º 64 = 0

46. x2 + y2 º 16x + 39 = 0 47. x2 º 4y2 º 20x º 64y º 172 = 0x2 º y2 º 9 = 0 4x2 + y2 º 80x + 16y + 400 = 0

48. x2 º 2x + 4 + y2 º 10 = 0 49. 4x2 º y2 º 8x + 6y º 9 = 02y2 º x + 3 = 0 2x2 º 3y2 + 4x + 18y º 43 = 0

50. 10x2 º 25y2 º 100x = º160 51. x2 º y º 4 = 0y2 º 2x + 16 = 0 x2 + 3y2 º 4y º 10 = 0

SYSTEMS OF THREE EQUATIONS Find the points, if any, that the graphs of all

three equations have in common.

52. x2 + y2 + 8x + 7 = 0 53. x2 + y2 º 8 = 0x2 + y2 + 4x + 4y º 5 = 0 x2 + y2 º 3x + y = 0x2 + y2 = 1 2x2 + 2y2 º 5x º 10 = 0

54. x2 + 3y2 = 16 55. x2 + y2 º 4x º 4y = 263x2 + y2 = 16 x2 + y2 º 4x = 54y = ºx y = 3x º 8

56. CRITICAL THINKING Suppose a line intersects a circle whose center is at theorigin, and the line passes through the origin. If you know one of the points ofintersection, how do you know what the other point of intersection is withoutsolving the system algebraically?

57. LOGICAL REASONING Sketch examples to illustrate the different numbers ofpoints of intersection that a circle and an ellipse can have if both are centered atthe origin.

58. LAW ENFORCEMENT Suppose a car is traveling down the highway at aconstant rate of 60 miles per hour. It passes a police car parked at the side of theroad. To catch up to the car, the police officer accelerates at a constant rate. Thedistance d (in miles) the police car has traveled as a function of time t (in hours)since the other car has passed it is given by d = 3600t2. Write and solve a system ofequations to calculate how long it takes the police car to catch up to the other car.

59. COMMUNICATIONS The range of a radio station is bounded by a circlegiven by the following equation:

x2 + y2 º 1620 = 0

A straight highway can be modeled by the following equation:

y = º}

13

}x + 30

Find the length of the highway that lies within the range of the radio station.

POLICE OFFICER

The duties of apolice officer vary. An offi-cer in a large city is oftenassigned to a specific typeof duty, while an officer in a small community usuallyperforms a variety of tasks.

CAREER LINK


INT

ERNET

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FOCUS ON

CAREERS

F G H I J K L M


60. BUS BOUNDARY To be eligible to ride the school bus to East HighSchool, a student must live at least 1 mile from the school. How long is theportion of Clark Street for which the residents are not eligible to ride the schoolbus? (Use a coordinate plane in which the school is at (0, 0) and each unitrepresents one mile.)

61. NAVIGATION LORAN (Long-Distance Radio Navigation) uses synchronizedpulses sent out by pairs of transmitting stations. By calculating the difference in thetimes of arrival of the pulses from two stations, the LORAN equipment on a shiplocates the ship on a hyperbola. By doing the same thing with a second pair of stations, LORAN locates the ship at the intersection of two hyperbolas. SupposeLORAN equipment indicates that a ship’s location is the point of intersection of the graphs in the following system:

xy º 24 = 0x2 º 25y2 + 100 = 0

Find the ship’s location given that it is north and east of the origin.

62. HYPERBOLIC MIRROR In a hyperbolicmirror, light rays directed to one focus will bereflected to the other focus. The mirror shown at the right has the following equation:

}

3x6

2

} º }

6

y

4

2

} = 1

At which point on the mirror will light from thepoint (0, 8) be reflected to the focus at (º10, 0)?

63. EARTHQUAKES An earthquake occurred in Peru on April 18, 1993. Use the following information to approximate the location of the epicenter.

c Source: U.S. Department of the Interior Geological Survey

Location 1: (Cayambe, Ecuador) The epicenter was 1300 kilometers away.

Location 2: (Cocohabamba, Bolivia, 1200 kilometers east and 1900 kilometerssouth of Cayambe) The epicenter was 1300 kilometers away.

Location 3: (Cerro El Oso, Venezuela, 1100 kilometers east and 1000 kilometersnorth of Cayambe) The epicenter was 2500 kilometers away.

64. MULTIPLE CHOICE How many points of intersection do the equations x2 + y2 = 6 and 2x2 + 4y2 = 7 have?

¡A 0 ¡B 1 ¡C 2 ¡D 3 ¡E 4

65. MULTIPLE CHOICE Which of the following is a point of intersection of thegraphs of 25x2 + 36y2 º 900 = 0 and º2x2 + y + 5 = 0?

¡A (º5, 0) ¡B (0, 5) ¡C (2, 5) ¡D (1, 5) ¡E (0, º5)

66. CRITICAL THINKING Write equations for three different conics that allintersect at the point (º4, 6).

2 mi

5 mi

1 mi

East High SchoolState St.

Main St.

Clark St.

NO P Q(0, 8)

(210, 0) (10, 0)

TestPreparation

★★Challenge

HOMEWORK HELP

Visit our Web sitewww.mcdougallittell.comfor help with problemsolving in Exs. 60–62.

INT

ERNET

STUDENT HELP

R S T U V W X Y


EVALUATING EXPRESSIONS Evaluate the expression for the given value of x.

(Review 1.2 for 11.1)

67. 2x + 5 when x = 4 68. }

x

13} º 1 when x = 2

69. (º2)x º 1 when x = 5 70. }(º3

3

)x º 2} when x = 4

WRITING FUNCTIONS Write a polynomial function of least degree that has

real coefficients, the given zeros, and a leading coefficient of 1. (Review 6.7)

71. 3, º3, 1 72. 0, 2, 2, 4 73. 2i, º2i

74. 3 + i, 3 º i 75. 2, º1, º1 º i 76. º2, º3, i, i

GRAPHING Graph the function. Then state the domain and range. (Review 7.5)

77. ƒ(x) = Ï2wxw+w 3w 78. ƒ(x) = 5Ïxwºw 8w 79. ƒ(x) = º(x + 4)1/2 + 2

80. ƒ(x) = º3Ï3

xw+w 1w 81. ƒ(x) = Ï3

4wxw+w 1w + 2 82. ƒ(x) = 5(x º 1)1/3

CLASSIFYING CONICS Classify the conic section. (Review 10.6)

83. 3x2 + y2 + 2x + 2y = 0 84. 4x2 º y2 º 8x + 4y º 9 = 0

85. x2 + 6x º 2y + 13 = 0 86. x2 + y2 º 2x + 6y + 9 = 0

Write an equation for the conic section. (Lesson 10.6)

1. Circle with center at (º3, º5) and radius 8

2. Ellipse with vertices at (º7, 2) and (6, 2) and foci at (4, 2) and (º5, 2)


4. Hyperbola with foci at (2, º1) and (2, 8) and vertices at (2, 3) and (2, 4)

Classify the conic section. (Lesson 10.6)

5. x2 + 4y2 º 8x + 3y + 12 = 0 6. º3x2 º 3y2 + 6x + 4y + 1 = 0

7. º2y2 + x + 5y + 26 = 0 8. º6x2 + 4y2 + 2x + 9 = 0

Find the points of intersection, if any, of the graphs in the system. (Lesson 10.7)

9. 3x2 º 4x º y + 2 = 0 10. ºx2 + y2 + 4x º 6y + 4 = 0y = º5x + 4 x2 + y2 º 4x º 6y + 12 = 0

11. x2 + y2 + 4y º 12 = 0 12. y2 º 6x º 2y º 3 = 0x2 º 16y2 º 64y º 80 = 0 2y2 º 4y + x + 6 = 0

13. SEISMOLOGY A seismograph records the epicenter of an earthquake50 miles away. A second seismograph, 50 miles west and 35 miles north of thefirst, records the epicenter as being 35 miles away. A third seismograph, 80 milesdue west of the first, records the epicenter 30 miles away. Where was theearthquake’s epicenter in relation to the first seismograph? (Lesson 10.7)

QUIZ 3 Self-Test for Lessons 10.6 and 10.7

MIXED REVIEW

Z [ \ ] ^ _ ` aExtension

C H A P T E R 1 0

Chapter 10 Extension 639

Find the eccentricity

of a conic section.

. To write equations for

real-life conics, such as the

moon’s orbit in Example 3.


GOAL


Eccentricity of Conic SectionsSome ellipses are more oval thanothers. In an ellipse that is nearlycircular, the ratio c:a is close to 0. Ina more oval ellipse, c:a is close to 1.This ratio is called the of the ellipse. Every conic has aneccentricity e associated with it.

Finding Eccentricity

Find the eccentricity of the conic section described by the equation.

a. (x + 2)2 = 4(y º 1) b. 25(x + 2)2 º 36(y º 1)2 = 900

SOLUTION

a. This equation describes a parabola. By definition, the eccentricity is e = 1.

b. This equation describes a hyperbola with a = Ï3w6w = 6, b = Ï2w5w = 5, and

c = Ïaw2w+w bw2w = Ï6w1w. The eccentricity is e = }ac

} = ≈ 1.302.

Using Eccentricity to Write an Equation

Find an equation of the hyperbola with center (3, º5), vertex (9, º5), and e = 2.

SOLUTION

Use the form }(x º

a2

h)2

} º = 1. The vertex lies 9 º 3 = 6 units from the

center, so a = 6. Because e = }ac

} = 2, you know that }6c

} = 2, or c = 12. Therefore,

b2 = c2 º a2 = 144 º 36 = 108. The equation is }(x º

36

3)2

} º }( y

1

+

08

5)2

} = 1.

(y º k)2

}

b2

E X A M P L E 2

Ï6w1w}

6

E X A M P L E 1

eccentricity

Let c be the distance from each focus to the center of the conic section, and let a

be the distance from each vertex to the center.

• The eccentricity of an ellipse is e = }a

c}, and 0 < e < 1.

• The eccentricity of a hyperbola is e = }a

c}, and e > 1.

• The eccentricity of a parabola is e = 1.

• The eccentricity of a circle is e = 0.

ECCENTRICITY OF CONIC SECTIONSCONCEPT

SUMMARY

a

c

a

c

majoraxis

minoraxis

Earth, as seen from

the moon

b c d e f g h i


Using Eccentricity to Write a Model

The moon orbits Earth in an elliptical path with the center of Earth at one focus.The eccentricity of the orbit is e = 0.055 and the length of the major axis is about768,800 kilometers. Find an equation of the moon’s orbit.

SOLUTION

Let the major axis of the ellipse be horizontal. The equation of the orbit has the form

}

a

x2

2} + }

b

y2

2} = 1. Using the length of the major axis, you know that 2a = 768,800,

or a ≈ 384,400. Because e = }ac

}, you know that 0.055 = }384

c,400}, or c ≈ 21,142

and b = Ïaw2wºw cw2w = Ï3w8w4w,4w0w0w2wºw 2w1w,1w4w2w2w = Ï1w.4w7w 3w 1w0w11w ≈ 383,800. The

equation of the moon’s orbit is }384

x

,4

2

002} + = 1 where x and y are

measured in kilometers.

EXERCISES

Find the eccentricity of the conic section.

1. 3x2 º 5x + y + 20 = 0 2. 25(x º 3)2 + 9(y + 6)2 = 225

3. x2 + 16(y º 4)2 = 16 4. }(x º

8

3)2

} + = 8

5. }(x +

25

6)2

} º }(y

1

º

00

6)2

} = 1 6. }(x +

49

2)2

} + }( y +

16

2)2

} = 1

7. 4(x + 1)2 º 8(y º 2)2 = 16 8. (x º 4)2 º (y º 3)2 = 1

Write an equation of the conic section.

9. Ellipse with vertices at (º5, º1) and (5, º1), and e = 0.6

10. Ellipse with foci at (2, º4) and (2, 4), and e = 0.5

11. Ellipse with center at (2, 0), focus at (2, 2), and e = 0.25

12. Ellipse with center at (0, 6), vertex at (3, 6), and e = 0.1

13. Hyperbola with foci at (3, º7) and (3, 9), and e = 3

14. Hyperbola with vertices at (º10, 4) and (º2, 4), and e = 2.4

15. Hyperbola with center at (3, 2), vertex at (3, 5), and e = 1.9

16. Hyperbola with center at (º1, 2), focus at (4, 2), and e = 5

17. ASTRONOMY Mercury orbits the sun in an elliptical path with the center of the sun at one focus. The eccentricity of Mercury’s orbit is e = 0.2056. Thelength of the major axis of the orbit is 72 million miles. Find an equation ofMercury’s orbit.

18. ASTRONOMY Mars orbits the sun in an elliptical path with the center of thesun at one focus. The eccentricity of Mars’ orbit is e = 0.0932. The perihelion ofMars’ orbit is the point where the planet is closest to the sun. At the perihelion,Mars’ distance from the sun is 128.4 million miles. Find an equation of Mars’ orbit.

19. Writing Explain why the definition of eccentricity for ellipses and hyperbolasimplies that 0 < e < 1 for an ellipse and e > 1 for a hyperbola.

(y º 5)2

}8

y2

}

383,8002

E X A M P L E 3

b c d e f g h iWHY did you learn it?

Find the distance a medical helicopter must travel. (p. 593)

Find the diameter of a broken dish. (p. 591)

Design a city park. (p. 593)

Model a solar energy collector. (p. 597)

Model the region lit by a lighthouse. (p. 603)

Model the shape of an Australian football field. (p. 614)

Model the curved sides of a sculpture. (p. 617)

Classify mirrors in a Cassegrain telescope. (p. 627)

Find the epicenter of an earthquake. (p. 634)

Find the area of The Ellipse at the White House. (p. 611)

641

Chapter SummaryCHAPTER

10

WHAT did you learn?

Find the distance between two points. (10.1)

Find the midpoint of the line segment connecting two points. (10.1)

Use distance and midpoint formulas in real-life situations. (10.1)

Graph and write equations of conics.

• parabolas (10.2, 10.6)

• circles (10.3, 10.6)

• ellipses (10.4, 10.6)

• hyperbolas (10.5, 10.6)

Classify a conic using its equation. (10.6)

Solve systems of quadratic equations. (10.7)

Use conics to solve real-life problems. (10.2–10.7)

How does Chapter 10 fit into the BIGGER PICTURE of algebra?

In Chapter 5 you studied parabolas as graphs of quadratic functions, and in Chapter 9you studied hyperbolas as graphs of rational functions. In a previous course youstudied circles, and possibly ellipses, in the context of geometry. In Chapter 10 youstudied all four conic sections (parabolas, hyperbolas, circles, and ellipses) as

graphs of equations of the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.

The conic sections are an important part of your study of algebra and geometrybecause they have many different real-life applications.

How did you makeand use a dictionaryof graphs?

Here is an example of one entryfor your dictionary of graphs,following the Study Strategyon page 588.

STUDY STRATEGY

Dictionary of GraphsCircle with center at (h, k) and radius r

Equation: (x º h)2+ (y º k)2

= r 2

r(h, k) (x, y)

y

x

b c d e f g h i


Chapter ReviewCHAPTER

10

• distance formula, p. 589

• midpoint formula, p. 590

• focus, p. 595, 609, 615

• directrix, p. 595

• circle, p. 601

• center, p. 601, 609, 615

• radius, p. 601

• equation of a circle, p. 601

• ellipse, p. 609

• vertex, p. 609, 615

• major axis, p. 609

• co-vertex, p. 609

• minor axis, p. 609

• equation of an ellipse, p. 609


• transverse axis, p. 615

• equation of a hyperbola, p. 615

• conic sections, p. 623

• general second-degreeequation, p. 626

• discriminant, p. 626

VOCABULARY

10.1 THE DISTANCE AND MIDPOINT FORMULAS

Let A = (º2, 4) and B = (2, º3).

Distance between A and B = Ï(xw2wºw xw1)w2w+w (wy2w ºw yw1)w2w= Ï(2w ºw (wºw2w))w2w+w (wºw3w ºw 4w)2w= Ï1w6w +w 4w9w = Ï6w5w ≈ 8.06

Midpoint of ABÆ

= MS}x1 +

2

x2}, }

y1 +

2

y2}D = S}(º2)

2

+ 2}, }

4 +

2

(º3)}D = S0, }

21

}D

Examples onpp. 589–591


segment connecting the two points.

1. (º2, º3), (4, 2) 2. (º5, 4), (10, º3) 3. (0, 0), (º4, 4) 4. (º2, 0), (0, º8)

Identify the focus and directrix of the parabola. Then draw the parabola.

5. x2 = 4y 6. x2 = º2y 7. 6x + y2 = 0 8. y2 º 12x = 0

Write the equation of the parabola with the given characteristic and vertex (0, 0).

9. focus: (4, 0) 10. focus: (0, º3) 11. directrix: y = º2 12. directrix: x = 1

10.2 PARABOLAS

The parabola with equation y2 = 8x has vertex (0, 0) and a horizontal axis of symmetry. It opens to the right. Note that y2 = 4px = 8x, so p = 2. The focus is ( p, 0) = (2, 0), and the directrix is x = ºp = º2.

The parabola with equation x2 = º8y has vertex(0, 0) and a vertical axis of symmetry. It opens down. Note that x2 = 4py = º8y, so p = º2. The focus is (0, p) = (0, º2), and the directrix is y = ºp = 2.


EXAMPLES

EXAMPLES

j kA

B

llM s0, d1

2

j kl(2, 0)

x 5 22

l j kmn(0, 22)

y 5 2

o p q r s t u v10.3 CIRCLES

The circle with equation x2 + y2 = 9 has center at (0, 0) and radius r = Ï9w = 3.

Four points on the circle are (3, 0), (0, 3), (º3, 0), and (0, º3).


Graph the equation.

13. x2 + y2 = 16 14. x2 + y2 = 64 15. x2 + y2 = 6 16. 3x2 + 3y2 = 363

Write the standard form of the equation of the circle that has the given radius

or passes through the given point and whose center is the origin.

17. radius: 5 18. radius: Ï1w0w 19. point: (º2, 3) 20. point: (1, 8)

10.4 ELLIPSES

The ellipse with equation }

x9

2

} + }

y

4

2

} = 1

has a horizontal major axis because 9 > 4.

Since Ï9w = 3, the vertices are at (º3, 0) and (3, 0).

Since Ï4w = 2, the co-vertices are at (0, º2) and (0, 2).

Since 9 º 4 = 5, the foci are at (ºÏ5w, 0) and (Ï5w, 0).


Graph the equation.

21. 4x2 + 81y2 = 324 22. º9x2 º 4y2 = º36 23. 49x2 + 36y2 = 1764


24. Vertex: (0, 5), Co-vertex: (1, 0) 25. Vertex: (4, 0), Focus: (º3, 0)

EXAMPLE

EXAMPLE j kww

10.5 HYPERBOLAS

The hyperbola with equation }

y

4

2

} º }

x9

2

} = 1 has

a vertical transverse axis because the y2-term is positive.Since Ï4w = 2, vertices are (0, º2) and (0, 2). Since 4 + 9 = 13, foci are (0, ºÏ1w3w) and (0, Ï1w3w).

Asymptotes are y = }

23

}x and y = º}

23

}x.


EXAMPLE

j kww

Chapter Review 643

j klw

o p q r s t u v


10.6 GRAPHING AND CLASSIFYING CONICSExamples onpp. 623–627

Classify the conic section and write its equation in standard form. Then graph

the equation.

32. x2 + 8x º 8y + 16 = 0 33. x2 + y2 º 10x + 2y º 74 = 0

34. 9x2 + y2 + 72x º 2y + 136 = 0 35. y2 º 4x2 º 18y º 8x + 76 = 0

10.7 SOLVING QUADRATIC SYSTEMS

You can solve systems of quadratic equations algebraically.

y2 º 2x º 10y + 31 = 0

x º y + 2 = 0 Solve the second equation for y : y = x + 2.

(x + 2)2 º 2x º 10(x + 2) + 31 = 0 Substitute into the first equation.

x2 º 8x + 15 = 0, so x = 3 or x = 5. Simplify and solve.

The points of intersection of the graphs of the system are (3, 5) and (5, 7).


You can use the discriminant B2 º 4AC to classify a conic.

For the equation x2 + y2 º 6x + 2y + 6 = 0, the discriminant is B2 º 4AC = 02 º 4(1)(1) = º4. Because B2 º 4AC < 0, B = 0, and A = C, the equation represents a circle.

To graph the circle, complete the square as follows.

x2 + y2 º 6x + 2y + 6 = 0

(x2 º 6x + 9) + (y2 + 2y + 1) = º6 + 9 + 1

(x º 3)2 + (y + 1)2 = 4

The center of the circle is at (h, k) = (3, º1) and r = Ï4w = 2.

EXAMPLE

EXAMPLE

Graph the hyperbola.

26. }

1x0

2

0} º }

6

y

4

2

} = 1 27. 16y2 º 9x2 = 144 28. y2 º 4x2 = 4

Write an equation of the hyperbola with the given foci and vertices.

29. Foci: (0, º3), (0, 3) 30. Foci: (0, º4), (0, 4) 31. Foci: (º5, 0), (5, 0)Vertices: (0, º1), (0, 1) Vertices: (0, º2), (0, 2) Vertices: (º3, 0), (3, 0)


36. x2 + y2 º 18x + 24y + 200 = 0 37. 5x2 + 3x º 8y + 2 = 04x + 3y = 0 3x + y º 6 = 0

38. 4x2 + y2 º 48x º 2y + 129 = 0 39. 9x2 º 16y2 + 18x + 153 = 0 x2 + y2 º 2x º 2y º 7 = 0 9x2 + 16y2 + 18x º 135 = 0

10.5 continued

lw(3, 21)

j k

o p q r s t u v

Chapter Test 645

Chapter TestCHAPTER

10


segment connecting the two points.

1. (1, 9), (5, 3) 2. (º8, 3), (4, 7) 3. (º4, º2), (3, 10)

4. (º11, º5), (º3, 7) 5. (º1, 6), (2, 8) 6. (3, º2), (4, 9)

Graph the equation.

7. x2 + y2 = 36 8. y2 = 16x 9. 9y2 º 81x2 = 729

10. 25x2 + 9y2 = 225 11. (x º 4)2 = y + 7 12. (x º 3)2 + (y + 2)2 = 1

13. }

(x +

4

6)2

} + }(y º

1

7)2

} = 1 14. }

(x º

16

4)2

} º }(y +

16

4)2

} = 1 15. }

(y +

4

2)2

} º }(x +

16

1)2

} = 1

Write an equation for the conic section.

16. Parabola with vertex at (0, 0) and directrix x = 5


18. Circle with center at (0, 0) and passing through (4, 6)

19. Circle with center at (º8, 3) and radius 5

20. Ellipse with center at (0, 0), vertex at (4, 0), and co-vertex at (0, 2)

21. Ellipse with vertices at (3, º5) and (3, º1) and foci at (3, º4) and (3, º2)

22. Hyperbola with vertices at (º7, 0) and (7, 0) and foci at (º9, 0) and (9, 0)

23. Hyperbola with vertex at (4, 2), focus at (4, 4), and center at (4, º1)

Classify the conic section and write its equation in standard form.

24. x2 + 4y2 º 2x º 3 = 0 25. 2x2 + 20x º y + 41 = 0 26. 5x2 º 3y2 º 30 = 0

27. x2 + y2 º 12x + 4y + 31 = 0 28. y2 º 8x º 4y + 4 = 0 29. ºx2 + y2 º 6x º 6y º 4 = 0

30. x2 º 8x + 4y + 16 = 0 31. 3x2 + 3y2 º 30x + 59 = 0 32. x2 + 2y2 º 8x + 7 = 0

33. 4x2 º y2 + 16x + 6y º 3 = 0 34. 3x2 + y2 º 4y + 3 = 0 35. x2 + y2 º 2x + 10y + 1 = 0


36. x2 + y2 = 64 37. x2 + y2 = 20 38. x2 = 8y

x º 2y = 17 x2 + 4y2 º 2x º 2 = 0 x2 = 2y + 12

39. ARCHITECTURE The Royal Albert Hall in London is nearly elliptical inshape, about 230 feet long and 200 feet wide. Write an equation for the shape ofthe hall, assuming its center is at (0, 0). Then graph the equation.

40. SEARCH TEAM A search team of three members splits to search an area inthe woods. Each member carries a family service radio with a circular range of 3 miles. They agree to communicate from their bases every hour. One membersets up base 2 miles north of the first member. Where should the other memberset up base to be as far east as possible but within range of communication?

o p q r s t u v


Chapter Standardized TestCHAPTER

10

1. MULTIPLE CHOICE What is the midpoint of theline segment connecting points (0, 0) and (º8, 2)?

¡A (º4, 1) ¡B (4, 1) ¡C (4, º1)

¡D (1, 4) ¡E (1, º4)

2. MULTIPLE CHOICE Which equation represents theperpendicular bisector of the line segment connectingpoints (º7, 1) and (9, 13)?

¡A y = º}

43

}x + }

235} ¡B y = }

34

}x + }

245}

¡C y = }

43

}x + }

235} ¡D y = }

43

}x + }

137}

¡E y = º}

43

}x + }

137}

3. MULTIPLE CHOICE Which equation is graphed?

¡A x2 + y2 = 8 ¡B x

2 º y2 = 8

¡C x2 + y2 = 16 ¡D 9x

2 + 9y2 = 576

¡E 9x2 º 9y

2 = 576

4. MULTIPLE CHOICE What is the standard form ofthe ellipse with center at (0, 0), vertex at (0, 9), and co-vertex at (4, 0)?

¡A }

x

9

2

} + }

y

4

2

} = 1 ¡B }

x

4

2

} + }

y

9

2

} = 1

¡C }

8x

1

2

} + }

1

y

6

2

} = 1 ¡D }

1x

6

2

} + }

8

y

1

2

} = 1

¡E }

x

2

2

} + }

y

3

2

} = 1

5. MULTIPLE CHOICE What is the focus of theparabola with equation 2x

2 = º120y?

¡A (0, 60) ¡B (0, 15) ¡C (0, º60)

¡D (0, 12) ¡E (0, º15)

6. MULTIPLE CHOICE What is the directrix of theparabola with equation y2 = 24x?

¡A x = 6 ¡B x = º6 ¡C x = 24

¡D y = 6 ¡E y = º6

7. MULTIPLE CHOICE Which graph represents the

equation }

2

y

5

2

} º }

x

9

2

} = 1?

¡A ¡B

¡C ¡D

¡E

8. MULTIPLE CHOICE What conic does the equationx

2 º 5x + 10y + 11 = 0 represent?

¡A circle ¡B ellipse

¡C hyperbola ¡D parabola

¡E none of the above

9. MULTIPLE CHOICE What point is the intersectionof the graphs of x2 + y2 = 41 and y = 3x º 7?

¡A (º4, º4) ¡B (4, 5) ¡C (5, 8)

¡D (3, 2) ¡E (º4, º19)

x yl lx yl lx yl lx yl lx yl l

TEST-TAKING STRATEGY During the test, do not worry excessively about how much time you

have left. Concentrate on the question in front of you.

x yzz

o p q r s t u v

Chapter Standardized Test 647

QUANTITATIVE COMPARISON In Exercises 10 and 11, choose the statement

that is true about the given quantities.

¡A The quantity in column A is greater.

¡B The quantity in column B is greater.

¡C The two quantities are equal.

¡D The relationship cannot be determined from the given information.

10.

11.

12. MULTI-STEP PROBLEM Let (0, 0) represent a water fountain located in a citypark. Each day Jane runs through the park along a path given by the equation x

2 + y2 º 200x º 52,500 = 0 where x and y are measured in meters.

a. Writing What type of conic is Jane’s path? How do you know?

b. Write the equation of the conic in standard form. Then graph the equation.

c. After her run, Jane walks to the water fountain. If Jane stops running at(º100, 150), how far must she walk for a drink of water?

13. MULTI-STEP PROBLEM The Mars Global Surveyor spacecraft followed anelliptical path with the center of Mars at one focus. The spacecraft’s initial orbithad a low point of 262 kilometers above the northern hemisphere and a highpoint of 54,026 kilometers above the southern hemisphere. c Source: NASA

a. Writing The radius of Mars is approximately 5400 kilometers. If (0, 0)represents the center of Mars and the positive y-axis represents north, what are the coordinates of the other focus of the orbit? How do you know?

b. Write an equation for the spacecraft’s initial orbit around Mars.

c. In February, 1999, the spacecraft reached a nearly circular orbit, 410 kilometersabove the surface of Mars. Write and graph an equation of the orbit.

14. MULTI-STEP PROBLEM Sara Peters is a mail carrier for a post office thatreceives mail for everyone living within a radius of 5 miles. Her route covers theportions of Anderson Road and Murphy Road that pass through this region.

a. Assume that the post office is located at the point (0, 0). Write an equation forthe circle that bounds the region where the mail is delivered.

b. Assuming Anderson Road follows one branch of a hyperbolic path given by

x2 º y2 º 4x º 23 = 0, graph Anderson Road and the circular region

where Sara delivers mail.

c. Writing If Sara begins delivery on Anderson Road at the point (º4, º3),where on Anderson Road does she end delivery? How do you know?

d. Sara finishes delivering on Anderson Road at the point where it intersects boththe circular boundary and Murphy Road. At the intersection, she beginsdelivering on Murphy Road which is a straight road that cuts through the centerof the circular region past the post office. Find the equation that representsMurphy Road. Where does Sara Peters end delivery on Murphy Road?

Column A Column B

Distance between (3, º2) and (º5, 7) Distance between (º8, º1) and (0, 8)

Discriminant of x2 + y2 º 6x + 1 = 0 Discriminant of 3x2 + y2 º 2y + 5 = 0

Documents

QUADRATIC RELATIONS AND CONIC SECTIONSsciannamath.weebly.com/.../2/...and_conic_sections.pdf · Chapter 10 is about conic sections. The four conic sections are parabolas, circles,