Find the equation of a hyperbola satisfying these conditions?

Question

Find the equation of a hyperbola satisfying the given conditions:

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

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

Answer 1

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.