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hyperbola

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hyperbola

Last Updated: March 11, 2008

- The set of all co-planar points whose difference of the distances from two fixed points (foci) are constant.

foci

foci

Center: (h, k)

conjugate axis

vertex

vertex

transverse axis

Co-vertices endpoints ofconjugate axis

vertex

transverse axis

Co-vertices endpoints ofconjugate axis

conjugate axis

vertex

Distance from center to vertex = a

Distance from center to co-vertex = b

Distance from center to foci = c

b

a

c2= a2 + b2

c

Length of transverse axis = 2a

Length of conjugate axis = 2b

The Latus Rectum (LR) is a chord passing through the focus that is perpendicular to an extended transverse axis.

The length

of the L.R. is

Distance from center to vertex = a

Distance from center to co-vertex = b

Distance from center to foci = c

Center: (-1, 5)

a = 4 in x direction

b = 7 in y direction

(-1, 12)

7

(-1, 5)

(-5, 5)

4

4

(3, 5)

7

(-1, -2)

Center: (-1, 5)

a = 4 b = 7

a2 + b2 = c2

(-1, 12)

42 + 72 = c2

16 + 49 = c2

65 = c2

7

foci

(-1, 5)

(-5, 5)

4

4

(3, 5)

7

(-1, -2)

Asymptotes

(-1, 12)

7

(-1, 5)

(-5, 5)

4

4

(3, 5)

7

(-1, -2)

Asymptotes

(-1, 12)

7

(-1, 5)

(-5, 5)

4

4

(3, 5)

7

(-1, -2)

Find theLength of the LR.

a = 4 b = 7 c =

(-1, 12)

7

(-1, 5)

(-5, 5)

4

4

(3, 5)

7

(-1, -2)

Center: (-1, 5)

Vertices: (-5, 5) (3, 5)

Co-Vertices: (-1, 12) (-1, -2)

Foci:

(-1, 12)

Length of Transverse axis: 8

Length of Conjugate axis: 14

7

(-1, 5)

Asymptotes

(-5, 5)

4

4

(3, 5)

7

(-1, -2)

Center: (-2, 3)

a = 6 in y direction

b = 3 in x direction

(-2, 9)

6

(-2, 3)

(-5, 3)

3

3

(1, 3)

6

(-2, -3)

Center: (-2, 3)

a = 6 b = 3

a2 + b2 = c2

(-2, 9)

62 + 32 = c2

36 + 9 = c2

45 = c2

6

foci

(-2, 3)

(-5, 3)

3

3

(1, 3)

6

(-2, -3)

Asymptotes

(-2, 9)

6

(-2, 3)

(-5, 3)

3

3

(1, 3)

6

(-2, -3)

Asymptotes

(-2, 9)

6

(-2, 3)

(-5, 3)

3

3

(1, 3)

6

(-2, -3)

Find theLength of the LR.

a = 6 b = 3 c =

(-2, 9)

6

(-2, 3)

(-5, 3)

3

3

(1, 3)

6

(-2, -3)

Center: (-2, 3)

Vertices: (-2, -3) (-2, 9)

Co-Vertices: (-5, 3) (1, 3)

Foci:

(-2, 9)

Length of Transverse axis: 12

6

Length of Conjugate axis: 6

(-2, 3)

Asymptotes

(-5, 3)

3

3

(1, 3)

6

(-2, -3)

4x2 + 8x - 9y2 + 54y - 53 = 168

(4x2 + 8x ) - (9y2 - 54y ) = 168 + 53

4(x2 + 2x + 12) - 9(y2 - 6y + 32) = 221 + 4 - 81

4(x + 1)2 - 9(y - 3)2 = 144

144

36

16

Center: (-1, 3)

a = 6 in x direction

(-1, 7)

b = 4 in y direction

4

(-1, 3)

6

6

(-7, 3)

(5, 3)

4

(-1, -1)

Center: (-1, 3)

a = 6 b = 4

foci

a2 + b2 = c2

(-1, 7)

62 + 42 = c2

36 + 16 = c2

4

52 = c2

(-1, 3)

6

6

(-7, 3)

(5, 3)

4

(-1, -1)

Asymptotes

(-1, 7)

4

(-1, 3)

6

6

(-7, 3)

(5, 3)

4

(-1, -1)

Asymptotes

(-1, 7)

4

(-1, 3)

6

6

(-7, 3)

(5, 3)

4

(-1, -1)

Find theLength of the LR.

a = 6 b = 4 c =

(-1, 7)

4

(-1, 3)

6

6

(-7, 3)

(5, 3)

4

(-1, -1)

Center: (-1, 3)

Vertices: (-7, 3) (5, 3)

Co-Vertices: (-1, 7) (-1, -1)

Foci:

(-1, 7)

Length of Transverse axis: 12

Length of Conjugate axis: 8

4

(-1, 3)

Asymptotes:

6

6

(-7, 3)

(5, 3)

4

(-1, -1)

4x2 + 16x - 9y2 + 72y - 5 = 87

4x2 + 16x - 9y2 + 72y = 87 + 5

4(x2 + 4x + 22) - 9(y2 - 8y + (-4)2) = 92 + 16-144

4(x + 2)2 - 9(y - 4)2 = -36

-36

-9

4

Center: (-2, 4)

a = 2 in y direction

b = 3 in x direction

(-2, 6)

2

(-2, 4)

3

3

(-5, 4)

(1, 4)

2

(-2, 2)

Center: (-1, 3)

a = 2 b = 3

a2 + b2 = c2

(-2, 6)

22 + 32 = c2

4 + 9 = c2

foci

13 = c2

2

(-2, 4)

3

3

(-5, 4)

(1, 4)

2

(-2, 2)

Asymptotes

(-2, 6)

2

(-2, 4)

3

3

(-5, 4)

(1, 4)

2

(-2, 2)

Asymptotes

(-2, 6)

2

(-2, 4)

3

3

(-5, 4)

(1, 4)

2

(-2, 2)

Find theLength of the LR.

a = 2 b = 3 c =

(-2, 6)

2

(-2, 4)

3

3

(-5, 4)

(1, 4)

2

(-2, 2)

Center: (-2, 4)

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

Co-Vertices: (-5, 4) (1, 4)

Foci:

(-2, 6)

Length of Transverse axis: 4

Length of Conjugate axis: 6

2

Asymptotes

(-2, 4)

3

3

(-5, 4)

(1, 4)

2

(-2, 2)

That's

All

Folks