Answer : $\large\frac{x^2}{4}-\frac{y^2}{5}=\frac{4}{9}$
Given eccentricity $'e'=\large\frac{3}{2}$
Coordinate of foci =$(\pm 2,0)$
It is clear that the hyperbola lies on the $x$-axis.
Hence its equation is $\large\frac{x^2}{a^2}-\frac{y^2}{b^2}$$=1$
$ae=2$
$\therefore a(\large\frac{3}{2})$$=2$
$a=\large\frac{4}{3}$
$a^2=\large\frac{16}{9}$
$b^2=a^2(e^2-1)$
$b^2=\large\frac{16}{9}(\frac{9}{4}$$-1)$
$\;\;\;\;=\large\frac{16}{9}\times \frac{5}{4}$
$\;\;\;\;=\large\frac{20}{9}$
Hence the equation of the hyperbola is $\large\frac{9x^2}{16}-\frac{9y^2}{20}$$=1$
(ie) $\large\frac{x^2}{4}-\frac{y^2}{5}=\frac{4}{9}$