Slide 1 / 181 Slide 2 / 181 - NJCTLcontent.njctl.org/courses/math/algebra-ii/conic-sections/conic... · 21.04.2015 · Slide 1 / 181 Slide 2 / 181 Algebra II Conic Sections 2015-04-21

Slide 1 / 181 Slide 2 / 181

Algebra II

Conic Sections

www.njctl.org

2015-04-21

Slide 3 / 181

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

Slide 4 / 181

Midpoint and DistanceFormula

Return to Tableof Contents

Slide 5 / 181

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)

Slide 6 / 181

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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Slide 7 / 181 Slide 7 (Answer) / 181

Slide 8 / 181

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

A (2,3)

B (3,5)

C (-2,-3)

D (-3,-5)

Slide 8 (Answer) / 181

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

Slide 9 / 181

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

Slide 10 / 181

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

Slide 11 / 181

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 = =

Slide 12 / 181


Slide 14 / 181

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

Slide 15 / 181

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

Slide 16 / 181

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

Slide 17 / 181

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

A

B

C

D

Slide 18 / 181

Introduction to Conic

Sections


Slide 19 / 181

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

Discussion Question: Which conic sections are functions?

Slide 20 / 181

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

Slide 21 / 181

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

Slide 22 / 181

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

The Ellipse

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

http://njc.tl/lv

Slide 23 / 181

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

The Parabola

Slide 24 / 181

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

This is the only cross section that intersects both cones.

The Hyperbola

Slide 25 / 181

Parabolas


Slide 26 / 181

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".

Slide 27 / 181

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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Slide 28 / 181

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

Slide 29 / 181

Slide 29 (Answer) / 181 Slide 30 / 181

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

Slide 31 / 181


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

Slide 32 / 181


Slide 34 / 181


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

Slide 35 / 181


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


Slide 37 / 181

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

Converting from General Form to Standard Form

Slide 38 / 181


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

Slide 39 / 181






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

Slide 45 / 181

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

Slide 46 / 181

Slide 47 / 181





Ans

wer

Slide 48 / 181

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

Slide 49 / 181

21 What value completes the square of

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



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


Ans

wer

1

Slide 50 / 181

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

Slide 51 / 181

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

Slide 52 / 181

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

Slide 53 / 181


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



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


Ans

wer

4

Slide 54 / 181

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

Slide 55 / 181


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

Slide 56 / 181

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.

Slide 57 / 181

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

Slide 58 / 181

Directrix

Focus

L1

L2

L1=L2

Eccentricity of a Parabola

All parabolas have an eccentricity of 1.

Slide 59 / 181

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

Slide 60 / 181

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.

Slide 61 / 181

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

Slide 62 / 181

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.

Slide 63 / 181

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

Slide 64 / 181 Slide 65 / 181Graph the equation from the last example.

Dire

ctrix

Focus Axis of Symmetry

Slide 65 (Answer) / 181Graph 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.

Slide 66 / 181Identify 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?

Slide 66 (Answer) / 181Identify 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.

Slide 67 / 181

Graph


Graph


Ans

wer

Directrix

Axi

s of

Sym

met

ry

Slide 68 / 181Identify 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 1: Convert the equation from general to standard form.

Slide 68 (Answer) / 181Identify 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 1: Convert the equation from general to standard form.


Ans

wer

Slide 69 / 181Step 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?

Slide 69 (Answer) / 181Step 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.

Slide 70 / 181

Graph


Graph


Ans

wer

Slide 71 / 181

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

Slide 72 / 181

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

Slide 73 / 181

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)

Slide 74 / 181

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

Slide 75 / 181

33 What is the focal distance in the following equation?


33 What is the focal distance in the following equation?


Ans

wer

Slide 76 / 181

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

Slide 77 / 181

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

Slide 78 / 181

36 What is the eccentricity of the following conic section?


36 What is the eccentricity of the following conic section?


Ans

wer

1

Slide 79 / 181















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

[This object is a pull tab]A

nsw

er

B

Slide 94 / 181

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

t

The 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.

Slide 95 / 181



Ans

wer

Slide 96 / 181

Circles


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Slide 97 / 181

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

Slide 98 / 181

Slide 99 / 181 Slide 100 / 181

Slide 101 / 181

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

Slide 102 / 181

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

Slide 103 / 181

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

Slide 104 / 181

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

A

B

C

D



A

B

C

D


Ans

wer

C


Slide 106 / 181


A

B

C

D



A

B

C

D


Ans

wer

D

Slide 107 / 181

58 What is eccentricity of a circle?


58 What is eccentricity of a circle?


Ans

wer 0

Slide 108 / 181

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

Slide 109 / 181

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:


Slide 110 / 181


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


Slide 111 / 181


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.


Slide 112 / 181 Slide 113 / 181

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.)


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

Slide 115 / 181

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

Slide 116 / 181

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

Slide 117 / 181



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

Slide 120 / 181

Ellipses


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Slide 121 / 181

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

Slide 122 / 181

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.

Slide 123 / 181

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.

Slide 124 / 181Parts 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.

Slide 125 / 181

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

Slide 126 / 181

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

Slide 127 / 181

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

Slide 128 / 181

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

Slide 129 / 181

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

Slide 130 / 181 Slide 131 / 181

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.)

Slide 132 / 181

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

Slide 133 / 181

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

Slide 134 / 181

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

Slide 135 / 181

72 Name one foci of

A

B

C

D


72 Name one foci of

A

B

C

D


Ans

wer

C

Slide 136 / 181

73 Name one foci of

A

B

C

D


73 Name one foci of

A

B

C

D


Ans

wer D

Slide 137 / 181

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

Slide 138 / 181

Slide 139 / 181 Slide 140 / 181

Slide 141 / 181

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

Slide 142 / 181

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

Slide 143 / 181

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

Slide 144 / 181

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

Slide 145 / 181

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


Slide 147 / 181

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

Slide 148 / 181

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

Slide 149 / 181



80 Convert the following ellipses to standard form.

A

B

C

D



A

B

C


Ans

wer

C

Slide 152 / 181


A

B

C

D



A

B

C

D [This object is a pull tab]

Ans

wer C

Slide 153 / 181


Hyperbolas


Slide 155 / 181

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.)

Slide 156 / 181

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.)

Slide 157 / 181

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

Slide 158 / 181To 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)

page1svg

Slide 159 / 181

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

Slide 160 / 181

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








Slide 168 / 181

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


Slide 170 / 181

Convert to standard form:

Slide 171 / 181

Slide 172 / 181

Recognizing Conic Sections

fromGeneral Form


Slide 173 / 181

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.

page1svg








Slide 181 / 181

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

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Slide 1 / 181 Slide 2 / 181 - NJCTLcontent.njctl.org/courses/math/algebra-ii/conic-sections/conic... · 21.04.2015 · Slide 1 / 181 Slide 2 / 181 Algebra II Conic Sections 2015-04-21