Question

Find the equation of the hyperbola satisfying the give conditions: Vertices $(0,\pm 3)$ foci $(0,\pm 5)$

Answer 1

It is given that, vertices $(0,\pm 3),$ foci $(0,\pm 5)$
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 $(0,\pm 3)\Rightarrow a=3$
Since the foci are $(0,\pm 5)\Rightarrow ae=c=5$
We know that $a^2+b^2=c^2$
$\therefore 3^2+b^2=5^2$
$\Rightarrow b^2=25-9=16$
Thus the equation of the hyperbola is $\frac{y^2}{9}-\frac{x^2}{16}=1$