Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

Section 8.3Ellipses

Parabola HyperbolaCircle Ellipse

Ellipse:

Besides having the two foci, an ellipse also has a major and minor axis, vertices at the end of the major axis and center point where the two axes cross.

Standard Equations for an EllipseMajor axis Parallel to x - axis

x 2 y 2

a 2 b 2+ = 1

Center = (0, 0)Vertices (a, 0), (- a, 0)

a(a,0)

V(- a, 0)V

F

(- c , 0)

F

(c, 0)

Foci (c, 0), (- c, 0)c2 = a2 - b2

Major Axis = 2a Minor Axis = 2b

(0, 0)

Minor Intercepts (0, b), (0, -b)

b

(0, b)

(0, - b)

a > b > 0

Standard Equations for an EllipseMajor axis parallel to y - axis

x 2 y 2

b 2 a 2+ = 1

Center = (0, 0)Vertices (0, a), (0, - a)

Foci (0, c), (0, - c)

Major Axis = 2aMinor Axis = 2b

Minor Intercepts (b, 0), (- b, 0)

a

(0,a)V

(0,- a,) V

b(-b,0)

(b,0)(0, 0)

(0,c)F

F (0,-c)c2 = a2 - b2

a > b > 0

a2 = 16 a = 4

b2 = 9 b = 3

c2 = a2 - b2 = 16 - 9 = 7c = 7

Minor intercepts = (0, 3) & (0,- 3)

Maj. Axis = 2·a = 2(4) = 8 Min. Axis = 2·b = 2(3) = 6

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EllipseSketch, Find Foci, Length of Minor and Major Axis For Center at the origin.

x2 y2 16 9

+ = 1- 4

4

- 3

3

- 7 7

Vertices = (4, 0) & (- 4, 0)

Foci = (7, 0) & (- 7, 0)

a2 = 81 a = 9

b2 = 16 b = 4

c2 = a2 - b2 = 81 - 16 = 65c = 65

Vertices = (0, 9) & (0, - 9) Minor intercepts = (4,0) & (- 4,0)

Maj. Axis = 2·a = 2(9) = 18 Min. Axis = 2·b = 2(4) = 8

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EllipseSketch, Find Foci, Length of Minor and Major Axis For Center at the origin.

x2 y2 16 81

+ = 1- 4

4

- 9

9

- 65

65

Foci = (0, 65) & (0, - 65)

Graph the Ellipse1616 22 yx

1161

22

yx

12 b

1b 4a162 a

Needs to be set equal to 1.

Vertices: (0,-4) and (0,4)

Minor Intercepts: (-1,0) and (1,0)

Find the equation of the ellipse Foci: (-1,0) and (1,0)

Vertices: (-3,0) and (3,0)

Therefore a = 3 and c = 1

222 bac 291 b

28 b28 b

189

22

yx

EllipseFind an equation of an ellipse in the form

x2 y2 a2 b2

+ = 1

1. When Major axis is on x-axis Major axis length = 32 Minor axis length = 30

Therefore, a = 32 ÷ 2 = 16a2 = 256

b = 30 ÷ 2 = 15 b2 = 225

x2 y2 256 225

+ = 1

2. Major axis on y-axis Major axis length = 16 Distance from Foci to Center = 7

Ellipse

Therefore, c = 7

Find an equation of an ellipse in the form

x2 y2 b2 a2

+ = 1

a = 16 ÷ 2 = 8 a2 = 64

c2 = a2 – b2 b2 = a2 – c2 = 64 – 49 = 15

x2 y2 15 64 + = 1

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Find the equation of the ellipse in the form below

if thee center is the origin.

x2 y2 a2 b2

+ = 1

a = 10 b = 6

a2 = 100b2 = 36

x2 y2 100 36

+ = 1

Translations

Ellipses translate just like circles and parabolas do…by using h and k in the standard equation.

1)()(

2

2

2

2

b

ky

a

hx

This is for a horizontal major axis, switch a and b for a vertical major axis…if your equation isn’t in this form you will need to complete the square to make it so…

Graph the ellipse

1

1

3

9

1 22

yx

Center: (-1,3)

92 a

Major axis parallel to x-axis12 b

3a 1b

Place a point 3 units right and left of center 8c

82 c

192 c

222 bac Place a point 1 unit above and below the center.

8.2c

Foci are about 2.8 units to the left and right of center.

Graph the ellipse0572324 22 yyxx

572324 22 yyxx

57284 22 yyxx

1116457121684 22 yyxx

8144 22 yx

1

8

1

2

4 22

yx

1

8

1

2

4 22

yx

82 a22 b

Major axis is parallel to the y-axis

Center is (-4,1)

Place 2 points 1.4 unit right and left of center

Place 2 points 2.8 units up and down from center

4.1b 8.2a

Write the equation of the ellipseFoci: (2,-2) and (4,-2)

Vertices: (0,-2) and (6,-2)

Center is halfway between the vertices so the point (3,-2)

We know a = 3 and c = 1291 b

28 b28 b

Plug into standard form:

1

2

2

2

2

b

ky

a

hx

1

8

2

9

3 22

yx

Write the equation of the ellipseMajor axis vertical with length of 6 and minor axis length of 4 centered at (1,-4)

62 a3a

42 b2b

1

9

4

4

1 22

yx

1

2

2

2

2

a

ky

b

hx