Question

A hyperbola with a tangent line at point $P$ shown. A ray of light from point $R$ reflects off the hyperbola at point $P$. If the point $R$ is moved around, then which of the following is/are correct?

• A. When the ray $RP$ points directly at $F_2$, then the reflected ray points directly at origin.
• B. When the ray $RP$ points directly at $F_2$, then the reflected ray points directly at $A$
• C. When the ray $RP$ points directly at $F_2$, then the reflected ray points directly at $F_1$
• D. When the ray $RP$ points directly at $F_2$, then the reflected ray points directly at $R$

Answer 1

The correct option is C When the ray $RP$ points directly at $F_2$, then the reflected ray points directly at $F_1$

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.
  

So according to this property When the ray $RP$ points directly at $F_2$, then due to reflection property of the Hyperbola the reflected ray points directly at $F_1$

Hence the correct option is $C$.