Equation of hyperbola 3

Question

Write an equation of the hyperbola given that the center is at (2,-3), the vertices are at (2,3) and (2,-9) and the foci are at (2, +- 2 sqrt. 10).

Answer 1

vertices:(2, 3) and (2, -9),foci:(2, +- 2 sqrt. 10), and center:(h, k) = (2, -3)

The y-coordinates are different in vertices, indicating a vertical axis and vertical hyperbola:
(y-k)²/a² -(x-h)²/b² = 1

Distance between vertices, 2a  = √(-9-3)²+(2-2)² = 12

a = 6

Distance between foci.

2c =√(2 sqrt. 10+2 sqrt. 10)²+(2-2)² =√(4 sqrt. 10)²= 4 sqrt. 10

c=2 sqrt. 10

Find the value of b:

b²= c²- a²

    =(2 sqrt. 10)²- 6²

     =40-36

       =4

b= 2

Equation of the hyperbola

(y-(-3)²/6² -(x- 2)²/2² = 1

(y+3)²/6² -(x- 2)²/2² = 1