Question

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

Answer 1

We have 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 $(0,\pm 5)\Rightarrow a=5$
And the foci are $(0,\pm 8)\Rightarrow ae=8$
We know that $b^2=a^2(e^2-1)=a^2{e}^2-a^2=64-25=39$
Thus the equation of the hyperbola is $\frac{y^2}{25}-\frac{x^2}{39}=1$