Ellipses

Date: ____________

Ellipses

Standard Equation of an Ellipse

Center at (0,0)

9.4 Ellipses

x2

a2

y2

b2+ = 1

y

x

(–a, 0) (a, 0) (0, b)

(0, –b)

O

Horizontal Major Axis Vertical Major Axis

Vertices

Co-Vertices

Vertices

Co-Vertices

Graph the ellipse. Find the vertices and co-vertices.

x

y x2

25

y2

9+ = 1

(–5, 0) (5, 0)

(0, 3)

(0,-3)

a2 = 25

a = ±5b2 = 9

b = ±3

Vertices: (–5, 0) and (5,0)

Co-vertices:(0, 3) and (0,-3)

Horizontal Major Axis

Graph the ellipse. Find the vertices and co-vertices.

x

y x2

9

y2

25+ = 1

(–3, 0) (3, 0)

(0, 5)

(0,-5)

a2 = 9

a = ±3

b2 = 25

b = ±5

Vertices: (0,5) and (0,-5)

Co-vertices: (-3,0) and (3,0)

Vertical Major Axis

Translated Ellipses

Standard Equation of an Ellipse

Center at (h,k)

9.4 Ellipses

(x – h)2

a2

(y – k)2

b2+ = 1

y

x(h–a, 0)

(h+a, 0) (0, k+b)

(0, k–b)

(h,k)

Graph the ellipse

x

y

a2 = 36

a = ±6b2 = 16

b = ±4

(x – 2)2

36

(y + 5)2

16+ = 1

Center = (2,-5)

Vertices: (8,-5) and (-4,-5)

Co-vertices: (2,-1) and (2,-9)

Horizontal Major Axis

Graph the ellipse

x

y

a2 = 25

a = ±5b2 = 81

b = ±9

(x + 3)2

25

(y + 1)2

81+ = 1

Center = (-3,-1)

Vertices: (-3,8) and (-3,-10)

Co-vertices: (-8,-1) and (2,-1)

Vertical Major Axis

Write the equation of the ellipse in standard form. Graph. Find the center, vertices, and co-vertices.

4x2 + 25y2 = 100

100x2

25

y2

4+ = 1

x

y

a2 = 25

a = ±5b2 = 4

b = ±2

Center = (0,0)

Vertices: (-5,0) and (5,0)

Co-vertices: (0,2) and (0,-2)

Write the equation of the ellipse in standard form. Graph. Find the center, vertices, and co-vertices.

25x2 + 3y2 = 75

75x2

3y2

25+ = 1

x

y

a2 = 3

a ≈ ±1.73b2 = 25

b = ±5

Center = (0,0)

Vertices: (0,5) and (0,-5)

Co-vertices: (-1.73,0) and (1.73,0)

Write the equation of the ellipse in standard form. Graph. Find the center, vertices, and co-vertices.

x2 + 9y2 – 4x + 54y + 49 = 0

x2 – 4x + 9y2 + 54y = -49

4x2 – 4x + ____ + 9(y2 + 6y + ___) = -499 +4 +81

(x – 2)2 + 9(y + 3)2 = 36

36(x – 2)2

36

(y + 3)2

4+ = 1

(x – 2)2

36

(y + 3)2

4+ = 1

x

y

a2 = 36

a = ±6b2 = 4

b = ±2

Center = (2,-3)

Vertices: (-4,-3) and (8,-3)

Co-vertices: (2,-1) and (2,-5)

Write the equation of the ellipse in standard form. Graph. Find the center, vertices, and co-vertices.

4x2 + y2 +24x – 4y + 36 = 0

4x2 + 24x + y2 – 4y = -36

94(x2 + 6x + ____) + y2 – 4y + ___ = -364 +36 + 4

4(x + 3)2 + (y – 2)2 = 4

4(x + 3)2

1

(y – 2)2

4+ = 1

x

y

a2 = 1

a = ±1b2 = 4

b = ±2

Center = (-3,2)

Vertices: (-3,4) and (-3,0)

Co-vertices: (-4,2) and (-2,2)

(x + 3)2

1

(y – 2)2

4+ = 1

Write the equation of the ellipse in standard form. Graph. Find the center, vertices, and co-vertices.

4x2 + 9y2 – 16x +18y – 11 = 0

4x2 – 16x + 9y2 + 18y = 11

44(x2 – 4x + ____) + 9(y2 + 2y + ___) =111 +16+9

4(x – 2)2 + 9(y + 1)2 = 36

36(x – 2)2

9

(y + 1)2

4+ = 1

x

y

a2 = 9

a = ±3b2 = 4

b = ±2

Center = (2,-1)

Vertices: (5,-1) and (-1,-1)

Co-vertices: (2,1) and (2,-3)

(x – 2)2

9

(y + 1)2

4+ = 1

9.4 Ellipses

An ellipse is the set of all points P in a plane such that the sum of the distances from P to two fixed points, F1 and F2, called the foci, is a constant.

P

F1 F2

F1P + F2P = 2a

PP

2a

9.4 EllipsesHorizontal Major Axis:

a2 > b2

a2 – b2 = c2

x2

a2

y2

b2+ = 1

F1(–c, 0) F2 (c, 0)

y

x

(–a, 0) (a, 0) (0, b)

(0, –b)

O

length of major axis: 2alength of minor axis: 2b

9.4 Ellipses

F2(0, –c)

F1 (0, c)

y

x

(0, –b)

(0, b)

(a, 0)(–a, 0)

O

Vertical Major Axis:

b2 > a2

b2 – a2 = c2

x2

a2

y2

b2+ = 1

length of major axis: 2blength of minor axis: 2a

Find the foci.

x

y x2

25

y2

9+ = 1

25 – 9 = c2

16 = c2

±4 = c(–4, 0)

(4, 0)

Find the foci.

x

y x2

9

y2

25+ = 1

25 – 9 = c2

16 = c2

±4 = c

(0,4)

(0,-4)

Find the foci.

x

y x2

100

y2

36+ = 1

100 – 36 = c2

64 = c2

±8 = c(–8, 0) (8, 0)

Find the foci.

x

y

(x – 4)2

16

(y – 3)2

25+ = 1

25 – 16 = c2

9 = c2

±3 = c

(4, 6)

(4, 0)

Find the foci.

x

y

(x + 1)2

4

(y + 2)2

16+ = 1

16 – 4 = c2

12 = c2

±3.5 ≈ c (-1,1.5)

(-1,-5.5)