Question

Find the equation of hyperbola whose focus is $(a,0)$, directrix is $2x-y+a=0$ and eccentricity $=\frac{4}{3}$

Answer 1

Given that : $e=\frac{4}{3},S(a,0)$ and equation of directrix is $2x-y+a=0$

Let a point be $P(x,y)$, such that
$SP=ePM$, where $PM$ is perpendicular distance from $P(x,y)$ to directrix

$\Rightarrow \sqrt{(x-a{)}^2+(y-0{)}^2}=\frac{4}{3}\times \left|\frac{2x-y+a}{\sqrt{2^2+(-1{)}^2}}\right|$

$\Rightarrow \sqrt{(x-a{)}^2+y^2}=\frac{4}{3\sqrt{5}}|2x-y+a|$

$\Rightarrow (x-a{)}^2+y^2=\frac{16}{45}(2x-y+a{)}^2$
$\Rightarrow 45(x^2-2ax+a^2+y^2)=16(4x^2+y^2+a^2-4xy-2ay+4ax)$
$\Rightarrow 45x^2-90ax+45a^2+45y^2=64x^2+16y^2+16a^2-64xy-32ay+64ax$
$\Rightarrow 19x^2-29y^2-64xy+154ax-32ay-29a^2=0$

$\therefore$ Equation of hyperbola is
$19x^2-64xy-29y^2+154ax-32ay-29a^2=0$