Transcript
Page 1: Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

Section 8.3Ellipses

Parabola HyperbolaCircle Ellipse

Page 2: Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

Ellipse:

Page 3: Section 8.3 Ellipses Parabola Hyperbola Circle 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.

Page 4: Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

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

Page 5: Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

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

Page 6: Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

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)

Page 7: Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

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)

Page 8: Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

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)

Page 9: Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

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

Page 10: Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

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

Page 11: Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

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

Page 12: Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

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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

Page 13: Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

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…

Page 14: Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

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.

Page 15: Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

Graph the ellipse0572324 22 yyxx

572324 22 yyxx

57284 22 yyxx

1116457121684 22 yyxx

8144 22 yx

1

8

1

2

4 22

yx

Page 16: Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

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

Page 17: Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

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

Page 18: Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

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


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