Question

Find the equation of the hyperbola satisfying the given conditions,

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

Answer 1

The vertices are $(0,\pm 5)$ which lie on y - axis.
So the equation of the hyperbola in standard form is, $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$
$\therefore \text{The vertices}(0,\pm a)\text{is}(0,\pm 5)\Rightarrow \text{foci}(0,\pm ae)\text{is}(0,\pm 8)\Rightarrow ae=8\text{Now}ae=8\Rightarrow e=\frac{8}{a}\Rightarrow e=\frac{8}{5}\text{We know that b}=a\sqrt{e^2-1}\Rightarrow e=5\sqrt{\frac{64}{25}-1}=5\frac{\sqrt{39}}{5}=\sqrt{39}\text{Thus required equation of hyperbola is},\frac{y^2}{(5{)}^2}-\frac{x^2}{(\sqrt{39}{)}^2}=1\Rightarrow \frac{y^2}{25}-\frac{x^2}{39}=1.$