# Find the equation of the hyperbola satisfying the give conditions: Vertices (0, ±5), foci (0, ±8)

Question:

Find the equation of the hyperbola satisfying the give conditions: Vertices $(0, \pm 5)$, foci $(0, \pm 8)$

Solution:

Vertices $(0, \pm 5)$, foci $(0, \pm 8)$

Here, the vertices are on the $y$-axis.

Therefore, the equation of the hyperbola is of the form $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$.

Since the vertices are $(0, \pm 5), a=5$.

Since the foci are $(0, \pm 8), c=8$.

We know that $a^{2}+b^{2}=c^{2}$.

$\therefore 5^{2}+b^{2}=8^{2}$

$b^{2}=64-25=39$

Thus, the equation of the hyperbola is $\frac{y^{2}}{25}-\frac{x^{2}}{39}=1$.