# Find the equation of a hyperbola satisfying these conditions?

Find the equation of a hyperbola satisfying the given conditions:

vertices at (0, 3) and (0, -3)

foci at (0, 5) and (0, -5)

The vertices are (0, 3), (0, - 3) and foci (0, 5), (0, - 5)

The x coordinate of vertices and foci are same, so the required hyperbola is vertical.

Standard form of vertical hyperbola is [(y - k)2]/a2 - [(x - h)2]/b2 = 1

Where a is semi transverse axis and b is semi conjugate axis.

Center is (h, k),vertices are  (h, k+a) and (h, k-a), foci (h, k+c) and (h, k-c) and c is the distance from the center to each focus.

c = √(a2 + b2).

Find the center of the hyperbola.

Vertices (0, 3) and (0, - 3) = (h, k+a) and (h, k-a)

Therefore, h = 0

k + a = 3 ---> (1)

k - a = - 3 ---> (2)

Add the equations (1) and (2).

2k = 0

k = 0

Center (h, k) = (0, 0).

Find the values of a, b.

Substitute the k value in equation (1).

0 + a = 3

a = 3

Foci (h, k+c) and (h, k-c) = (0, 5) and (0, - 5)

k + c = 5 ---> (3)

Substitute the k value in equation (3).

0 + c = 5

c = 5

c = √(a2 + b2)

5 = √(32+ b2)

25 = 9 + b2

16 = b2

b = 4

Substitute a, b and (h, k) in standard form.

[(y - 0)2]/32 - [(x - 0)2]/52 = 1

The hyperbola equation is y2/9 - x2/25 = 1.