Question

Find the equation of the hyperbola satisfying the given conditions,

Vertices $(\pm 2,0),foci(\pm 3,0)$

Answer 1

The vertices are $(\pm 2,0)$ which lie on x - axis
So the equation of hyperbola in standard form is, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
$\therefore \text{The vertices}(\pm a,0)\text{is}(\pm 2,0)\Rightarrow a=2\text{Foci}(\pm ae,0)\text{is}(\pm 3,0)\Rightarrow ae=3\text{Now}ae=3\Rightarrow e=\frac{3}{a}\Rightarrow e=\frac{3}{2}\text{We know that}b=a\sqrt{e^2-1}\Rightarrow b=2\sqrt{\frac{9}{4}-1}=2\frac{\sqrt{5}}{2}=\sqrt{5}$
Thus required equation of hyperbola is,
$\frac{x^2}{(2{)}^2}-\frac{y^2}{(\sqrt{5}{)}^2}=1\Rightarrow \frac{x^2}{4}-\frac{y^2}{5}=1$