Question

Find the equation of the hyperbola satisfying the give conditions: Vertices $(\pm 2,0)$ foci $(\pm 3,0)$

Answer 1

Vertices $(\pm 2,0)$ foci $(\pm 3,0)$
Clearly the vertices are on the $x$-axis.
Therefore, the equation of the hyperbola is of the form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
Since the vertices are $(\pm 2,0)$

$\Rightarrow a=2$
and foci are $(\pm 3,o)\Rightarrow ae=3$
We know that $b^2=a^2(e^2-1)=a^2{e}^2-a^2=9-4=5$
Thus the equation of the hyperbola is $\frac{x^2}{4}-\frac{y^2}{5}=1$