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√10).

Answer 1

use what you know about hyperbolas.

Since the center and vertices are bot at x=2, the axis is vertical. So, given the center, we have

(y+3)^2/a^2 - (x-2)^2/b^2 = 1

However, you have garbled the foci. Since the center is at y = -3, the foci cannot be symmetric about the x-axis.

Fix that, and then, knowing that
a = 6
and c = the real distance to the foci,
and b^2 = c^2-a^2

then you can write the equation.