Question

Find the equation of the hyperbola centered at the origin that has a vertex at (5, 0) and a focus at (9, 0).

Answer 1

Refer to the diagram shown below.

The vertex is at (a,0), with a = 5.
The focus is at (c,0), with c = 9.
Therefore
b² = c² - a²  = 81 - 25 = 56

The equation of the hyperbola is
$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} =1, \\ or \\ \frac{x^{2}}{25} - \frac{y^{2}}{56}=1$

Answer:
$\frac{x^{2}}{25} - \frac{y^{2}}{56} =1$