Question

A ray emanating from the point $(5,0)$ is incident on the hyperbola $9x^2-16y^2=144$ at the point $P$ with abscissa $8$. Find the equation of the reflected ray after first reflection and point $P$ lies in first quadrant.

• A. $33x-13y+153=0$
• B. $43x-14y+173=0$
• C. $53x-15y+173=0$
• D. $63x-16y+193=0$

Answer 1

The correct option is A $33x-13y+153=0$

Given hyperbola is $9x^2-16y^2=144$. This equation can be rewritten as $\frac{x^2}{16}-\frac{y^2}{9}=1$ ......(1)

Since $x$ coordinate of $P$ is $8$. Let $y$ coordinate of $P$ be $\alpha$.

$(8,\alpha )$ lies on (1)

$\frac{64}{16}-\frac{{\alpha }^2}{9}=1$

${\alpha }^2=27$ ---> $\alpha =33$ ($P$ lies in first quadrant)

Hence coordinate of point $P$ is $(8,33)$.

Equation of reflected ray passing through $P(8,33)$ and $S^{\prime }(-5,0)$

Its equation is $y-33=\left[\frac{(0-33)}{(-5-8)}\right](x-8)$

or $13y-393=33x-243$

or $33x-13y+153=0$.

Hence option (A) is correct.