Slide 1 / 181content.njctl.org/courses/math/algebra-ii/conic-sections/conic-sections-3/conic...Apr 21, 2015 · Slide 27 / 181 The equation of a parabola can be written in two forms:

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Algebra II

Conic Sections

www.njctl.org

2015-04-21




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Table of Contents

Review of Midpoint and Distance Formulas

Parabolas

Circles

Ellipses

Hyperbolas

Recognizing Conic Sections from General Form

Introduction to Conic Sections

click on the topic to go to that section

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Midpoint and DistanceFormula

Return to Tableof Contents

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A (-3, 6)

B (5, 2)

What is the midpoint of segment AB?

Look at this segment - the midpoint is halfway. To find the coordinates of the midpoint, find the average of the x-values and the average of the y-values.

(x,y)

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The Midpoint FormulaGiven points A(x1,y1) and B (x2,y2), the midpoint of AB is

Examples: Find the midpoint of the segment with the given endpoints.

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1 Find the midpoint of K(1,8) & L(5,2).

A (2,3)

B (3,5)

C (-2,-3)

D (-3,-5)


1 Find the midpoint of K(1,8) & L(5,2).

A (2,3)

B (3,5)

C (-2,-3)

D (-3,-5)[This object is a pull tab]

Ans

wer

B

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2 Find the midpoint of H(-4 , 8) & L(6, 10).

A (5,9)

B (-1,9)

C (1,9)

D (5,1)


2 Find the midpoint of H(-4 , 8) & L(6, 10).

A (5,9)

B (-1,9)

C (1,9)

D (5,1)[This object is a pull tab]

Ans

wer

C

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3 If the midpoint of a segment is (4,9) and one endpoint is (-3,10), find the other endpoint.

A (-10,8)

B (11,8)

C (-10,11)

D (.5,9.5)


3 If the midpoint of a segment is (4,9) and one endpoint is (-3,10), find the other endpoint.

A (-10,8)

B (11,8)

C (-10,11)

D (.5,9.5)

[This object is a pull tab]

Ans

wer

B

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A (1, 4)

B (7, -2)C (1,-2)

How far apart are points A and B?

The Distance Formula is derived from the Pythagorean Theorem, a2 + b2 = c2.

In this example,

AC2 + CB2=AB2

62+62 = AB2

72 = AB2

AB = =

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4 What is the distance between (2, 4) and (-1, 8)?


4 What is the distance between (2, 4) and (-1, 8)?


Ans

wer

5

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5 What is the distance between (0, 7) and (5, -5)?


5 What is the distance between (0, 7) and (5, -5)?


Ans

wer

13

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6 Given A( 4, 5) and B(x, 1) and AB=5, find all of the possible values of x.

A -7

B -5

C -3

D -1

E 0

F 1

G 3

H 5

I 7

J 9


6 Given A( 4, 5) and B(x, 1) and AB=5, find all of the possible values of x.

A -7

B -5

C -3

D -1

E 0

F 1

G 3

H 5

I 7

J 9[This object is a pull tab]

Ans

wer

1 and 7

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7 If the distance between (4,5) and (x,-2) is 10, what are the possible values of x?

A

B

C

D

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Introduction to Conic

Sections


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Conic Sections are created by intersecting a set of double cones with a plane.

Discussion Question: Which conic sections are functions?

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Click on the title below to take you to a webpage for more background information about conic sections:

"The Occurrence of the Conics", by Dr. Jill Britton

Click the link below for a YouTube video that demonstrates the cutting of the cones.

SalMathGuy Conics Video

More Info About Conics

http://britton.disted.camosun.bc.ca/jbconics.htm

http://njc.tl/lv

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A Circle comes from cutting parallel to the "base".

The term base is misleading because like lines and planes, conic sections continue on forever.

The Circle

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An Ellipse comes from cutting skew to the "base".

The Ellipse

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A Parabola comes from intersecting the cone with a plane that is parallel to a side of the cone.

The Parabola

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A Hyperbola comes from cutting the cones perpendicular to the "bases".

This is the only cross section that intersects both cones.

The Hyperbola

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Parabolas


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This is the graph of y = x2.Complete the table below:

x y-3-2-10123

Discuss the patterns that you observe.

A graph that has this shape is called a parabola.

y = x2 is the "parent function".

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The equation of a parabola can be written in two forms: y = ax2 + bx + c (the General Form)

y = a(x - h)2 + k (Standard Form)where (h,k) is the vertex. This is also called Vertex Form.

Example: Name the vertex of each equation: A) y= -3(x - 4)2 + 5

B) y= 2(x + 7)2 + 2 C) y= (x -3)2


The equation of a parabola can be written in two forms: y = ax2 + bx + c (the General Form)

y = a(x - h)2 + k (Standard Form)where (h,k) is the vertex. This is also called Vertex Form.

Example: Name the vertex of each equation: A) y= -3(x - 4)2 + 5

B) y= 2(x + 7)2 + 2 C) y= (x -3)2


Ans

wer

A) (4,5)

B) (-7,2)

C) (3,0)

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Match each equation to its parabola - drag the number of the graph to its equation.

f(x) = (x - 3)² - 2

g(x) = -2(x + 1)²

h(x) = 2 / 3 (x + 5)² - 7

1 2

3

2

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9 What is the vertex of ?

A (3, 2)

B (-3, -2)

C (2, 3)

D (-2, -3)



A (3, 2)

B (-3, -2)

C (2, 3)

D (-2, -3)


Ans

wer

D

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A (2, -3)

B (-3, -2)

C (2, 3)

D (-2, -3)



A (2, -3)

B (-3, -2)

C (2, 3)

D (-2, -3)


Ans

wer

A

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A (-3, 2)

B (-3, -2)

C (2, 3)

D (-2, -3)



A (-3, 2)

B (-3, -2)

C (2, 3)

D (-2, -3)


Ans

wer

A

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A (3, 2)

B (-3, -2)

C (2, 3)

D (-2, -3)



A (3, 2)

B (-3, -2)

C (2, 3)

D (-2, -3)


Ans

wer

B

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Fill in the blank to complete the square:Half of 6 is 3, 32 = 9

Converting from General Form to Standard Form

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y = x2 - 8x + 5

y = (x2 - 8x +___) + 5 - _____

What number completes the square in the parenthesis above?



y = x2 - 8x + 5

y = (x2 - 8x +___) + 5 - _____

What number completes the square in the parenthesis above?


Ans

wer y = (x2 - 8x + 16 ) + 5 - 16

y = (x - 4)2 - 11

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18

A (4, 5)

B (-4, 5)

C (-5, 4)

D (5, 4)

What is the vertex of x = y2 - 10y + 29?


18

A (4, 5)

B (-4, 5)

C (-5, 4)

D (5, 4)

What is the vertex of x = y2 - 10y + 29?


Ans

wer

A

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19 What is the vertex of y= x2 - 8x +21?

A (4, 5)

B (-4, 5)

C (-5, 4)

D (5, 4)


19 What is the vertex of y= x2 - 8x +21?

A (4, 5)

B (-4, 5)

C (-5, 4)

D (5, 4)


Ans

wer

A

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Ans

wer

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20 What should be factored out of

x = (4y2 - 8y + ___)+ 9 - ___ ?


20 What should be factored out of

x = (4y2 - 8y + ___)+ 9 - ___ ?


Ans

wer

4

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21 What value completes the square of

x = 4(y2 - 2y + ___)+ 9 - ___ ?



x = 4(y2 - 2y + ___)+ 9 - ___ ?


Ans

wer

1

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22 What value should follow "+ 9" in

x = 4(y2 - 2y + ___) + 9 ___ ?


22 What value should follow "+ 9" in

x = 4(y2 - 2y + ___) + 9 ___ ?


Ans

wer

-4

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23 Which is the correct standard form of

x = (4y2 - 8y + ___)+ 9 - ___ ?

A x = 4(y - 1)2 + 8 B x = 4(y + 1)2 + 8

C x = 4(y - 1)2 + 5

D x = 4(y + 1)2 + 5



x = (4y2 - 8y + ___)+ 9 - ___ ?

A x = 4(y - 1)2 + 8 B x = 4(y + 1)2 + 8

C x = 4(y - 1)2 + 5

D x = 4(y + 1)2 + 5 [This object is a pull tab]

Ans

wer

C

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24 What should be factored out of y = (-5x2 - 20x + ___)+ 7 - ___ ?


24 What should be factored out of y = (-5x2 - 20x + ___)+ 7 - ___ ?


Ans

wer

-5

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y = -5(x2 + 4x + ___)+ 7 - ___ ?click to reveal



y = -5(x2 + 4x + ___)+ 7 - ___ ?click to reveal


Ans

wer

4

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26 What value should follow "+7" in

y = -5(x2 + 4x + ___)+ 7 ___ ?click to reveal


26 What value should follow "+7" in

y = -5(x2 + 4x + ___)+ 7 ___ ?click to reveal


Ans

wer

20

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y = (-5x2 - 20x + ___)+ 7 - ___ ?

A y = -5(x - 2)2 + 3

B y = -5(x + 2)2 + 27

C y = -5(x - 2)2 -13

D y = -5(x - 2)2 + 27



y = (-5x2 - 20x + ___)+ 7 - ___ ?

A y = -5(x - 2)2 + 3

B y = -5(x + 2)2 + 27

C y = -5(x - 2)2 -13

D y = -5(x - 2)2 + 27


Ans

wer

B

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A parabola is a locus* of points equidistant from a fixed point, the focus, and a fixed line, the directrix.

*locus is just a fancy word for set.

Geometric Definition

Every parabola is symmetric with respect to a line through the focus and perpendicular to the directrix. The vertex of the parabola is the "turning point" and is on the axis of symmetry.

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Focus and Directrix of a Parabola

Axis of Symmetry

Directrix

Focus

Every point on the parabola is the same distance from the directrix and the focus.

L1

L2

L1=L2

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Directrix

Focus

L1

L2

L1=L2

Eccentricity of a Parabola

All parabolas have an eccentricity of 1.

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Parts of a ParabolaParts are the same for all parabolas, regardless of the direction in which they open.

Directrix

Axis of Symmetry

Vertex

FocusVertex

Focus

Directrix

Axis of Symmetry

x=ay2+by+cy=ax2+bx+c

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Compare the graphs below: What makes the graph more "narrow" or "wide"?

y = x2

y = 2x2

y = .5x2


Compare the graphs below: What makes the graph more "narrow" or "wide"?

y = x2

y = 2x2

y = .5x2


Ans

wer

As the absolute value of the coefficient of x2 increases, the parabola becomes more narrow. As it decreases, the parabola becomes wider.

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28 Which of the parabolas below are narrower than their parent functions?

A

B

C

D


28 Which of the parabolas below are narrower than their parent functions?

A

B

C

D


Ans

wer

C and D

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Focal Distance

To calculate:

focal distance =

The distance from the vertex to the focus is 1.

The distance from the vertex to the directrix is 1.

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Genera l Form y= ax2 + bx + c x= ay2 +by + c

S tandard Form y= a(x - h)2 +k x= a (y - k)2 + h

Opens a>0 opens upa<0 opens down

a>0 opens to the righta<0 opens to the le ft

Axis of Symmetry x = h y = k

Vertex (h , k) (h , k)

Foca l Dis tance

Directrix

Focus

Eccentricity 1 1

Parabola Summary

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Graph the equation from the last example.

Dire

ctrix

Focus Axis of Symmetry


Graph the equation from the last example.

Dire

ctrix

Focus Axis of Symmetry

Teac

her N

otes

[This object is a teacher notes pull tab]

Suggest that in order to get the right curve that students find one or two more points to plot and reflect.

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Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity?




Ans

wer

Focal distance=

The parabola opens up and has an eccentricity of 1.

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Graph


Graph


Ans

wer

Directrix

Axi

s of

Sym

met

ry

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Step 1: Convert the equation from general to standard form.



Step 1: Convert the equation from general to standard form.


Ans

wer

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Step 2: Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity?


Step 2: Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity?


Ans

wer

Focal distance=

The parabola opens to the left and has an eccentricity of 1.

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Graph


Graph


Ans

wer

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29 Given the following equation, which direction does it open?

A Up

B Down

C Left

D Right


29 Given the following equation, which direction does it open?

A Up

B Down

C Left

D Right


Ans

wer D, Right

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30 How does the following equation compare to the parent function

A Is narrower B Is widerC Is the same width


30 How does the following equation compare to the parent function

A Is narrower B Is widerC Is the same width


Ans

wer C, Wider

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31 Where is the vertex for the following equation?

A (-3 , 4)

B (3 , 4)

C (4 , 3)

D (4 , -3)


31 Where is the vertex for the following equation?

A (-3 , 4)

B (3 , 4)

C (4 , 3)

D (4 , -3)


Ans

wer C, (4,3)

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32 What is the equation of the axis of symmetry for the following equation?

A y = 3

B y = -3

C x = 4

D x = -4


32 What is the equation of the axis of symmetry for the following equation?

A y = 3

B y = -3

C x = 4

D x = -4


Ans

wer A, y=3

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33 What is the focal distance in the following equation?


33 What is the focal distance in the following equation?


Ans

wer

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34 What is the equation of the directrix for the following equation?

A y = 2

B y = -4

C x = 3

D x = -5


34 What is the equation of the directrix for the following equation?

A y = 2

B y = -4

C x = 3

D x = -5[This object is a pull tab]

Ans

wer

C

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35 Where is the focus for the following equation?

A (-3 , 5)

B (3 , 5)

C (5 , 3)

D (5 , -3)


35 Where is the focus for the following equation?

A (-3 , 5)

B (3 , 5)

C (5 , 3)

D (5 , -3)


Ans

wer C

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36 What is the eccentricity of the following conic section?


36 What is the eccentricity of the following conic section?


Ans

wer

1

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51 What is the equation of the parabola with vertex (2,3) and directrix y = 4?

A y = 4(x - 2)2 + 3

y = -1/4(x - 2)2 + 3

x = 4(y - 2)2 + 3

x = 1/4(y - 2)2 + 3

B

C

D


51 What is the equation of the parabola with vertex (2,3) and directrix y = 4?

A y = 4(x - 2)2 + 3

y = -1/4(x - 2)2 + 3

x = 4(y - 2)2 + 3

x = 1/4(y - 2)2 + 3

B

C

D


Ans

wer

B

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The St. Louis Arch is 630 feet tall and 630 feet wide at the base. Write an equation to represent the shape of the arch.

Challenge Problem

Answer on next page...


The St. Louis Arch is 630 feet tall and 630 feet wide at the base. Write an equation to represent the shape of the arch.

Challenge Problem

Answer on next page...


Hin

tThe vertex will be (315,630). Let (0,0) and (0,630) be points on the base of the parabola. Substitute either point and the vertex into the standard form equation and solve for a.

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Ans

wer

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Circles


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A circle is a locus of points in a plane that are equidistant from a given point.

Radius

Center (h,k)

(x,y)

The distance from the center to a point on the circle is

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52 Write the equation of the circle with center (5 , 2) and radius 6

A

B

C

D


52 Write the equation of the circle with center (5 , 2) and radius 6

A

B

C

D[This object is a pull tab]

Ans

wer C

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53 Write the equation of the circle with center (-5,0) and radius 7

A

B

C

D


53 Write the equation of the circle with center (-5,0) and radius 7

A

B

C


Ans

wer

B

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54 Write the equation of the circle with center (-2,1) and radius

A

B

C

D


54 Write the equation of the circle with center (-2,1) and radius

A

B

C


Ans

wer D

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55 What is the center and radius of the following equation?

A

B

C

D



A

B

C

D


Ans

wer

C

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A

B

C

D



A

B

C

D


Ans

wer

D

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58 What is eccentricity of a circle?


58 What is eccentricity of a circle?


Ans

wer 0

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Write the equations for each part of this unfortunate snowman.


Write the equations for each part of this unfortunate snowman.


Ans

wer

The base: (x - 3)2 + (y - 3)2 = 9

The torso: (x - 3)2 + (y - 8.5)2 = 6.25

The head: (x - 10)2 + (y - 2)2 = 4

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Write the equation of the circle that meets the following criteria:

Center (1 , -2) and passes through (4 , 6)

Since we know the center we only need to find the radius. The radius is the distance from the center to the point.

The equation of the circle is:


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Diameter with endpoints (4 , 7) and (-2 , -1).


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Center at (-5 , 6) and tangent to the y-axis.

"Tangent to the y-axis" means the circle only touches the y-axis at one point. Look at the graph.


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Write the equation of the circle in standard form that meets the following criteria:

Complete the square for the x's. (Remember, the y-term is 0y.)


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59 What is the equation of the circle that has a diameter with endpoints (0,0) and (16,12)?

A

B

C

D


59 What is the equation of the circle that has a diameter with endpoints (0,0) and (16,12)?

A

B

C

D


Ans

wer

A

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60 What is the equation of the circle with center (-3,5) that contains the point (1,3)?

A

B

C

D


60 What is the equation of the circle with center (-3,5) that contains the point (1,3)?

A

B

C

D


Ans

wer

C

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61 What is the equation of the circle with center (7,-3) and tangent to the x-axis?

A

B

C

D


61 What is the equation of the circle with center (7,-3) and tangent to the x-axis?

A

B

C

D


Ans

wer

D

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Challenge Question: What is the equation of a circle that passes through the three points (2,3), (2,-2), and (5,-3)?

Remember that the distance from the radius to the circle is the same for every radius. Let (x,y) be the center and use the distance formula twice.


Challenge Question: What is the equation of a circle that passes through the three points (2,3), (2,-2), and (5,-3)?

Remember that the distance from the radius to the circle is the same for every radius. Let (x,y) be the center and use the distance formula twice.


Ans

wer

(x - 4.5)2 + (y - .5)2 = 12.5

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Ellipses


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An ellipse is a focus of points in a plane that are each the same total distance from 2 fixed points, called the foci (plural of focus).

F1 F2

P2P1

For example, P1F1 + P1F2 = P2F1 + P2F2

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Eccentricity of an Ellipse0 < e < 1

The eccentricity of an ellipse is a number between 0 and 1. The more elongated the ellipse the closer the eccentricity is to 1. The closer an ellipse is to being a circle, the closer the eccentricity is to 0.

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A

D

B

C

O

The major axis, AB, is the segment through both foci whose endpoints are on the ellipse.The minor axis, CD, is perpendicular to the major axis through the center, O. The vertices of an ellipse are the endpoints of the major axis, points A and B. The co-vertices are the endpoints of the minor axis, points C and D.

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Parts of an Ellipse

Major axis

Maj

or a

xis

Minor axisMin

or a

xis

Vertex

Co-vertex

VertexCo-vertex

Vertex

Co-vertex

Focus

Horizontal ellipse Vertical ellipse

The length of the major axis is 2a.

The length of the minor axis is 2b.

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64 What letter or letters corresponds with ellipse's center?

A

B

C

D

E

F

G

A

D

B

C

FE G


64 What letter or letters corresponds with ellipse's center?

A

B

C

D

E

F

G

A

D

B

C

FE G


Ans

wer F

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65 What letter or letters corresponds with ellipse's foci?

A

B

C

D

E

A

D

B

C

FE G

FG


65 What letter or letters corresponds with ellipse's foci?

A

B

C

D

E

A

D

B

C

FE G

FG


Ans

wer E,G

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66 What letter or letters corresponds with ellipse's major axis?

A

B

C

D

E

F

G

H

I

ABCD G

E F

H

I


66 What letter or letters corresponds with ellipse's major axis?

A

B

C

D

E

F

G

H

I

ABCD G

E F

H

I


Ans

wer B

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67 Which choice best describes an ellipse's eccentricity?

A e = 0

B 0< e < 1

C e = 1

D e > 1


67 Which choice best describes an ellipse's eccentricity?

A e = 0

B 0< e < 1

C e = 1

D e > 1


Ans

wer B

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68 Which of the ellipses has the greater eccentricity?

AB A B


68 Which of the ellipses has the greater eccentricity?

AB A B


Ans

wer B

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a

ab

c

a

Finding the foci:

In this ellipse, a = 5 and b = 4, so c = 3.

The coordinates of the foci are (3-3,2) and (3+3,2) or (0,2) and (6,2)

(Note that in this case, a represents the hypotenuse of the triangle.)

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69 What is the center of

A (9 , 4)

B (5 , 6)

C (-5 , -6)

D (3 , 2)

?


69 What is the center of

A (9 , 4)

B (5 , 6)

C (-5 , -6)

D (3 , 2)

?


Ans

wer B

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70 How long is the major axis of

A 9

B 6

C 3

D 2

?


70 How long is the major axis of

A 9

B 6

C 3

D 2

?


Ans

wer B

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71 How long is the minor axis of

A 9

B 4

C 3

D 2


71 How long is the minor axis of

A 9

B 4

C 3

D 2


Ans

wer B

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72 Name one foci of

A

B

C

D


72 Name one foci of

A

B

C

D


Ans

wer

C

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73 Name one foci of

A

B

C

D


73 Name one foci of

A

B

C

D


Ans

wer D

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Graphing an Ellipse

· Find and graph the center · Find the length and direction of the major and minor axes· Draw the major and minor axes· Draw the ellipse

The center is (4 , -2)The major axis is 6 units and horizontalThe minor axis is 4 units and vertical

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74 Given that an ellipse has foci (4,1) and (-4,1) and major axis of length 10, what is the center of the ellipse?

A (8 , 2)

B (0 , 2)

C (0 , 1)

D (-8 , 1)


74 Given that an ellipse has foci (4,1) and (-4,1) and major axis of length 10, what is the center of the ellipse?

A (8 , 2)

B (0 , 2)

C (0 , 1)

D (-8 , 1)[This object is a pull tab]

Ans

wer C

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75 Given that an ellipse has foci (4,1) and (-4,1) and major axis of length 10, in which direction is the ellipse elongated?

A horizontally

B vertically

C obliquely

D it is not elongated


75 Given that an ellipse has foci (4,1) and (-4,1) and major axis of length 10, in which direction is the ellipse elongated?

A horizontally

B vertically

C obliquely

D it is not elongated[This object is a pull tab]

Ans

wer A

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76 Given that an ellipse has foci (4,1) and (-4,1) and major axis of length 10, how far is it from the center to an endpoint of the major axis?

A 10

B 100

C 5

D 25


76 Given that an ellipse has foci (4,1) and (-4,1) and major axis of length 10, how far is it from the center to an endpoint of the major axis?

A 10

B 100

C 5

D 25[This object is a pull tab]

Ans

wer C

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77 Given that an ellipse has foci (4,1) and (-4,1) and major axis of length 10, which equation would be used to find the distance from the center to an endpoint of the minor axis?

A

B

C

D


77 Given that an ellipse has foci (4,1) and (-4,1) and major axis of length 10, which equation would be used to find the distance from the center to an endpoint of the minor axis?

A

B

C

D [This object is a pull tab]

Ans

wer A

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78 Given that an ellipse has foci (4 , 1) and (-4 , 1) and major axis of length 10, find b.

A

B

C

D


78 Given that an ellipse has foci (4 , 1) and (-4 , 1) and major axis of length 10, find b.

A

B

C


Ans

wer D

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80 Given that an ellipse has foci (4,-4) and (4,2) and

minor axis of length 8, which is the equation of the

ellipse?


80 Given that an ellipse has foci (4,-4) and (4,2) and

minor axis of length 8, which is the equation of the

ellipse?


Ans

wer

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Steps for Converting the ellipse from General Form to Standard Form

· factor the x's and y's

· divide by the constant

· complete the square for x and/or y

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80 Convert the following ellipses to standard form.

A

B

C

D



A

B

C


Ans

wer

C

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A

B

C

D



A

B

C

D [This object is a pull tab]

Ans

wer C

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Slide 154 / 181

Hyperbolas


page1svg

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Like the ellipse, the hyperbola is a set of points at a given distance from two foci. In the case of the hyperbola, the absolute value of the difference of the distances from a point to the foci is constant.

a

F1

dc

b

|a - b| = |c - d|F2

(Don't worry so much about this definition - it is just to put things in perspective.)

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Vertex FocusFocus

Vertex Center

AsymptoteAsymptote

a ab

b

Horizontal Hyperbola

Equation:

Vertices: move a units to the left and right of the center

Foci: move c units to the left and right of the center, where

Asymptotes: slope = ± b/a (The asymptotes are lines that pass through the vertices of the rectangle between the vertices with length 2a and width 2b. An asymptote is a line that the graph approaches but never touches.)

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Vertex Focus

FocusVertex Center

Asym

ptot

eAsymptote

a

a

bb

Vertical Hyperbola

Equation:

Vertices: move a units up and down from the center

Foci: move c units up and down from the center, where

Asymptotes: slope = ± a/b

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To graph a hyperbola in standard form:· Find and graph the center· Plot points a right and left of the center, and b up and down for

horizontal, or b right and left, and a up and down for vertical· Make a rectangle through the four points from previous step· Draw asymptotes that contain the diagonals of the rectangle · Sketch the graph of the hyperbola

Center: (-1,2)

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Example: Graph

The cente r of the rectangle is ( -5 , 4 )

From the cente r move le ft/right 2

From the cente r move up/down 3

The hyperbola opens up and down

What are the slopes of the asymptotes?

How does this relate to a and b? Why?

Ans

wer

click

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Example: Graph

The cente r of the rectangle is ( 6 , 0 )

From the cente r move le ft/right 4

From the cente r move up/down 5

The hyperbola opens le ft and right

click

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90 What is the equation of a hyperbola that has vertices (±6,0) and foci (±10,0)?

A

B

C

D


90 What is the equation of a hyperbola that has vertices (±6,0) and foci (±10,0)?

A

B

C

D


Ans

wer

The hyperbola is horizontal.

a2 + b2 = c2, so 62 + b2 = 102

b = 8

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Convert to standard form:

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Recognizing Conic Sections

fromGeneral Form


page1svg

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General Form: ax2 + bx + cy2 + dy + e = 0

This form could represent any conic under the following conditions:

In a parabola, either a=0 or b=0.

ax2 + bx + dy +e =0

cy2 + dy + bx + e=0

In a circle, a=c and both a and c are positive. ax2 + bx + cy2 + dy + e = 0

In an ellipse, a and c are both positive, and a≠c. * ax2 + bx + cy2 + dy + e = 0

In a hyperbola, either a<0 and c>0 or a>0 and c<0.

ax2 + bx - cy2 + dy + e = 0

cy2 + dy - ax2 + bx + e = 0

* A circle is a special type of ellipse in which a = c.

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Eccentricity of Conic Sections

This picture depicts the comparative eccentricity of conic sections. Eccentricity (e) is a measure of "unroundness". A circle is round, so has e=0. For an ellipse, as the ellipse becomes more elongated, e increases from 0 to 1, not-including 1. A parabola has e=1, and for a hyperbola e>1.

Circle

e=0

Ellipse

0<e<1

Parabola

e = 1

Hyperbola

e > 1

Documents

Slide 1 / 181content.njctl.org/courses/math/algebra-ii/conic-sections/conic-sections-3/conic...Apr 21, 2015 · Slide 27 / 181 The equation of a parabola can be written in two forms: