# FIND THE EQUATION OF THE HYPERBOLA WITH THE GIVEN INFORMATION

## About "Find the Equation of the Hyperbola with the Given Information"

Find the Equation of the Hyperbola with the Given Information :

Here we are going to see some practice questions on find the equation of the hyperbola with the given information.

## Find the Equation of the hyperbola with the Given Information - Practice questions

Question 1 :

Find the equation of the hyperbola in each of the cases given below:

(i) foci (± 2, 0) , eccentricity = 3/2.

Solution :

F1 (2, 0) and F2 (-2, 0) and e = 3/2

From the given information, we know that the given ellipse is symmetric about x axis.

Midpoint of foci  =  Center of the ellipse

Center  =  (2 + (-2))/2, (0 + 0)/2  =  C (0, 0)

Distance between foci  =  √(x2 - x1)2 + (y2 - y1)2

=  √(2 + 2)2 + (0 - 0)2

=  √42 + 02

2ae  =  4

ae  =  2

e = 3/2

a (3/2) =  2

a  =  4/3 ==>  a2  =  16/9

b2  =  a2 (e2 - 1)

b2  =  (ae)2 - a2

b2  =  4 - (16/9)

b2  =  20/9

Equation of hyperbola :

(x2/a2) - (y2/b2)  =  1

(9x2/16) - (9y2/20)  =  1

(ii) Centre (2, 1) , one of the foci (8, 1) and corresponding directrix x = 4 .

Solution :

Distance between center and foci  = √(x2-x1)2 + (y2-y1)2

ae  =  √(8-2)2 + (1-1)2

ae  =  √36

ae  =  6 -----(1)

Equation of directrix x = 4

a/e  =  4 -----(2)

(1) x (2) ==>  a2  =  24

b2  =  a2 (e2 - 1)

b2  =  (ae)2 - a2

b2  =  36 - 24

b2  =  12

[(x-h)2/a2] - [(y-k)2/b2]  =  1

[(x-2)2/24] - [(y-1)2/12]  =  1

(iii) passing through (5,−2) and length of the transverse axis along x axis and of length 8 units.

Solution :

Length of transverse axis  =  8

2a  =  8

a  =  4

a2  =  16

(x2/a2) - (y2/b2)  =  1

The hyperbola passes through the point (5, -2).

(52/42) - (22/b2)  =  1

(25/16)  - (4/b2)  =  1

(25/16) - 1  =  (4/b2)

(4/b2)  =  9/16

64/9  =  b2

(x2/16) - (y2/(64/9))  =  1

(x2/16) - (9y2/64)  =  1  After having gone through the stuff given above, we hope that the students would have understood, "Find the Equation of the Hyperbola with the Given Information".

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