1

A hyperbola is the collection of points in the plane the difference of whose distances from two fixed points, called the foci, is a constant.

HYPERBOLA

2

3

Theorem Equation of a Hyberbola; Center at (0, 0); Foci at ( + c, 0); Vertices at ( + a, 0); Transverse Axis along the x-Axis

An equation of the hyperbola with center at (0, 0), foci at ( - c, 0) and (c, 0), and vertices at ( - a, 0) and (a, 0) is

x

a

y

bb c a

2

2

2

22 2 21 where

The transverse axis is the x-axis.

4

b c a2 2 2

x

a

y

b

2

2

2

2 1

5

Theorem Equation of a Hyberbola; Center at (0, 0); Foci at ( 0, + c); Vertices at (0, + a); Transverse Axis along the y-Axis

An equation of the hyperbola with center at (0, 0), foci at (0, - c) and (0, c), and vertices at (0, - a) and (0, a) is

y

a

x

bb c a

2

2

2

22 2 21 where

The transverse axis is the y-axis.

6

b c a2 2 2

y

a

x

b

2

2

2

2 1

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Theorem Asymptotes of a Hyperbola

The hyperbolax

a

y

b

2

2

2

2 1

has the two oblique asymptotes

y

bax y

bax and

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Theorem Asymptotes of a Hyperbola

The hyperbola y

a

x

b

2

2

2

2 1

has the two oblique asymptotes

y

abx y

abx and

10

Find an equation of a hyperbola with center at the origin, one focus at (0, 5) and one vertex at (0, -3). Determine the oblique asymptotes. Graph the equation by hand and using a graphing utility.

Center: (0, 0) Focus: (0, 5) = (0, c)

Vertex: (0, -3) = (0, -a)

Transverse axis is the y-axis, thus equation is of the form

y

a

x

b

2

2

2

2 1

11

y

a

x

b

2

2

2

2 1 a c2 29 25 ,

b c a2 2 2 = 25 - 9 = 16

y x2 2

9 161

Asymptotes: yab

x x 34

12

5 0 5

5

5

V (0, 3)

V (0, -3)

(4, 0)(-4, 0)

F(0, 5)

F(0, -5)

y x34

y x 34

y x2 2

9 161

13

Equation Foci Vertices Asymptotes

x h

a

y k

b

2

2

2

21

(h+c, k) (h+a, k) y kba

x h

Hyperbola with Transverse Axis Parallel to the x-Axis; Center at (h, k) where b2 = c2 - a2.

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Hyperbola with Transverse Axis Parallel to the y-Axis; Center at (h, k) where b2 = c2 - a2.

Equation Foci Vertices Asymptotes

y k

a

x h

b

2

2

2

21

(h, k+c) (h, k+a) y kab

x h

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Find the center, transverse axis, vertices, foci, and asymptotes of 4 16 8 16 02 2x x y y .

4 16 8 16 02 2x x y y

4 4 8 162 2x x y y

4 4 8 162 2x x y y _ _

42

42

82

162

4 4 4 8 16 16 16 162 2x x y y

4 2 4 162 2x y

x y

2

4

4

161

2 2

18

x y

24

416

12 2 x h

a

y k

b

2

2

2

2 1

Center: (h, k) = (-2, 4)

Transverse axis parallel to x-axis.

a b c a b2 2 2 2 24 16 4 16 20 , ,

a b c 2 4 2 5, , Vertices: (h + a, k) = (-2 + 2, 4) or (-4, 4) and (0, 4)

Foci: or

and

( , ) ( , )

( , ) ( , )

h c k

2 2 5 4

2 2 5 4 2 2 5 4

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Asymptotes: y kba

x h

y x 442

2( )

y x 4 2 2

a b c 2 4 2 5, , (h, k) = (-2, 4)

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10 0 10

10

C(-2,4)

V (-4, 4) V (0, 4)

F (2.47, 4)F (-6.47, 4)

(-2, 8)

(-2, 0)

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

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Sketch the curve represented by the equation:

091321849 22 yxyx

Exercise :

22

091321849 22 yxyx

0918429 22 yyxx

091164168419129 22 yyxx

Solution:Solution:

364419 22 yx

23

19

)4(

4

)1( 22

yx

13

)4(

2

)1(2

2

2

2

yx

Note:-To understand what this curve might look like, we have to work

towards a standard form. This is best accomplished by completing the

square in the x terms and in the y terms.

From this, we see that the curve is a hyperbola centered

at (1, 4). When y = 4 we have: 12

)1(2

2

x

24

• So,

• Thus, or

22 21 x

21-xor 21 x

3x 1x

Therefore, (3, 4) and 4,1 are both on the curve.

The asymptotes are the lines xy2

3 and xy

2

3

and they pass through the centre (1, 4).