Question

If a line passes through $(-5,0)$ and meet hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ at point $P$, then the reflected line passes through

• A. $(0,5)$
• B. $(0,10)$
• C. $(5,0)$
• D. $(0,-5)$

Answer 1

Answer: (C)

$(5,0)$

For given Hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$, $a=4$ and $b=3$

Eccentricity of hyperbola $e=\sqrt{1+\frac{b^2}{a^2}}$

Hence value of $e=\sqrt{1+\frac{9}{16}}=\frac{5}{4}$

We know that for a hyperbola there are two points of focus,. For any hyperbola with origin at $(0,0)$ and coordinate axes as major and minor axes, the two foci lie at $(ae,0)$ and $(-ae,0)$ respectively.

So Focus points of given hyperbola will be at $(4\times \frac{5}{4},0)$ and $(-4\times \frac{5}{4},0)$

$\to$ one focus will be at $(-5,0)$ and another focus will be at $(5,0)$

Every hyperbola has it's own reflection property. When a light ray following the path starting from one focus falls on Hyperbola, it will reflect off of the hyperbola directly away from the other focus.

Similarly, if a ray directed towards one focus will reflect off of the hyperbola towards the other focus.

From the given information we can understand that the point $(-5,0)$ is a focal point and line passing through it and falling on hyperbola at a point $P$ will reflect off of the hyperbola directly away from the other focus point $(5,0)$, because of its reflection property as explained above.

Hence we can say the reflected line passes through $(5,0)$ after falling off of the hyperbola. Correct option is $C$