A ray of light through 2 , 1 is reflected at a point P on the y - axis and then passes through the point 5 ,…

The reflected point of $\left(2,1\right)$ is $\left(-2,1\right).$

Equation of reflected ray

$y-1=\frac{3-1}{5+2}\left(x+2\right)$

$\Rightarrow y-1=\frac{2}{7}\left(x+2\right)$

$\Rightarrow 7y-7=2x+4$

$\Rightarrow 2x-7y+11=0$

Let the equation of other directrix is

$2x-7y+\lambda =0$

Distance of directrix from focus

$\frac{a}{e}-ae=\frac{8}{\sqrt{53}}$, where $e$ is the eccentricity

$\Rightarrow 3a-\frac{a}{3}=\frac{8}{\sqrt{53}}$ or $a=\frac{3}{\sqrt{53}}$

Distance from other focus $\frac{a}{e}+ae$

$=3a+\frac{a}{3}=\frac{10a}{3}=\frac{10}{3}\times \frac{3}{\sqrt{53}}=\frac{10}{\sqrt{53}}$

Distance between two directrix $=\frac{2a}{e}$

$=2\times 3\times \frac{3}{\sqrt{53}}=\frac{18}{\sqrt{53}}$

So, $\left|\frac{\lambda -11}{\sqrt{53}}\right|=\frac{18}{\sqrt{53}}$

$\Rightarrow \lambda -11=18$ or $-18$

$\Rightarrow 2x-7y-7=0$ or $2x-7y+29=0$