Question

Find the equation of the hyperbola whose vertices are $(0,\pm 3)$ and the foci are $(0,\pm 5).$

Answer 1

Since the vertices of the given hyperbola are of the form $(0,\pm a),$ it is a vertical hyperbola.

Let the required equation be $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1.$

Then, its vertices are $(0,\pm a).$

But, it is given that the vertices are $(0,\pm 3).$

$\therefore a=3.$

Let its foci be $(0,\pm c).$

But, it is given that the foci are $(0,\pm 5).$

$\therefore c=5.$

Now, $b^2=(c^2-a^2)=(5^2-3^2)=(25-9)=16.$

Thus, $a^2=3^2=9$ and $b^2=16.$

Hence, the required equation is $\frac{y^2}{9}-\frac{x^2}{16}=1.$