Translating Conic Sections

Section 10.6

Translating an Ellipse

Write an equation of an ellipse with center (-3, -2), vertical major axis of length 8 and minor axis of length 6.

(0, 4)

(0, -4)

(3, 0)(-3, 0)

(-3, -2)

1)()(

2

2

2

2

b

ky

a

hx

116

)2(

9

)3(

14

))2((

3

))3((

22

2

2

2

2

yx

yx

Translating an Ellipse

Write an equation of an ellipse with center (1, 8), horizontal major axis of length 16 and minor axis of length 10.

(0, 5)

(0, -5)

(8, 0)(-8, 0)

(1, 8)

1)()(

2

2

2

2

b

ky

a

hx

125

)8(

64

)1(

15

)8(

8

)1(

22

2

2

2

2

yx

yx

Translating a HyperbolaWrite an equation of a hyperbola with vertices (0, 1) and (6, 1) and foci (-1, 1) and (7, 1)

(6, 1)(7, 1)

(0, 1)(-1, 1)

2. Find the center, add a to lesser vertex

c = (7 – (-1))/2 = 4

1. Find the length of a, ½ the distance between the vertices

a = (6 – 0))/2 = 3

4. Find the length of b, by Pythagorean theorem.

3. Find the length of c, ½ the distance between the two foci

(3, 1)

Translating a HyperbolaRelationships between:1.Demonstrate how the location of the center of the hyperbola moves.2.Finding the length of c

Horizontal_Hyperbola.htmlVertical_Hyperbola.html

Translating a HyperbolaWrite an equation of a hyperbola with vertices (0, 1) and (6, 1) and foci (-1, 1) and (7, 1)

(6, 1)(7, 1)

(0, 1)(-1, 1)

1. a= 32. Center: (3, 1)3. c = 44. 42 = 32 + b2

16 = 9 + b2

7 = b2

c = (7 – (-1))/2 = 4

a = (6 – 0))/2 = 3

17

)1(

9

)3( 22

yx

(3, 1)

Translating a HyperbolaWrite an equation of a hyperbola with vertices (2, -1) and (2, 7) and foci (2, 10) and (2, -4)

(2, -4)

(2, 7)

(2, -1)

(2, 10)

1. b= 42. Center: (2, 3)3. c = 74. 72 = a2 + 42

49 = a2 + 16 33 = a2

c = (10 + 4)/2 = 7

b = (7 +1))/2 = 4

133

)2(

16

)3( 22

xy

(2, 3)

Assignment