Question

A ray emitting from the point $(-3,0)$ is incident on the ellipse $16x^2+25y^2=400$ at point $P$ with an ordinate $4.$ If the equation of the reflected ray after first reflection is $4x+3y=k,$ then the value of $k$ is

• A. $12$
• B. $-12$
• C. $9$
• D. $-9$

Answer 1

The correct option is A $12$

$16x^2+25y^2=400\Rightarrow \frac{x^2}{25}+\frac{y^2}{16}=1$
$a=5,b=4$
$\therefore c=\sqrt{25-16}=3$
Coordinates of foci are $(\pm c,0)$
i.e., $S(3,0)$ and $S^{\prime }(-3,0)$

We know that, ray emitting from focus $S^{\prime }$ after striking the ellipse at $P,$ will pass through other focus $S.$


Hence, point $S(3,0)$ lies on the given line $4x+3y=k.$
$\Rightarrow 4\left(3\right)+0=k\Rightarrow k=12$