A ray of light through (2, 1) is reflected at a point P on the y -...

The reflected point of $\left(2, 1\right)$ is \left(- 2, 1\right) .$</p><p><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/0187573e-6468-4151-bb74-e8f5ec5fdf83-image.png" style="width: 306px; height: 250px;" /></p><p>Equation of reflected ray</p><p>$y - 1 = \dfrac{3 - 1}{5 + 2}\left(x + 2\right)$</p><p>$⇒ y - 1 = \dfrac{2}{7}\left(x + 2\right)$</p><p>$⇒ 7y - 7 = 2x + 4$</p><p>$⇒ 2x - 7y + 11 = 0$</p><p>Let the equation of other directrix is</p><p>$2x - 7y + \lambda = 0$</p><p>Distance of directrix from focus</p><p>$\dfrac{a}{e} - ae = \dfrac{8}{\sqrt{53}}$, where$e$is the eccentricity</p><p>$⇒ 3a - \dfrac{a}{3} = \dfrac{8}{\sqrt{53}}$or$a = \dfrac{3}{\sqrt{53}}$</p><p>Distance from other focus$\dfrac{a}{e} + ae$</p><p>$= 3a + \dfrac{a}{3} = \dfrac{10a}{3} = \dfrac{10}{3} \times \dfrac{3}{\sqrt{53}} = \dfrac{10}{\sqrt{53}}$</p><p>Distance between two directrix$= \dfrac{2a}{e}$</p><p>$= 2 \times 3 \times \dfrac{3}{\sqrt{53}} = \dfrac{18}{\sqrt{53}}$</p><p>So, \left|\dfrac{\lambda - 11}{\sqrt{53}}\right| = \dfrac{18}{\sqrt{53}}$

$⇒ \lambda - 11 = 18$ or $- 18$

$⇒ 2x - 7y - 7 = 0$ or $2x - 7y + 29 = 0$