Question

A reflecting surface is represented by the equation $y=\frac{2L}{\pi }sin\left(\frac{\pi x}{L}\right),0\le x\le L$. A ray travelling horizontal becomes vertical after reflection. The co-ordinates of the point(s) on which this ray is incident.

• A. $(\frac{L}{4};\frac{\sqrt{2}L}{\lambda })$
• B. $(\frac{L}{3};\frac{\sqrt{3}L}{\pi })$
• C. $(\frac{3L}{4};\frac{\sqrt{2}L}{\lambda })$
• D. $(\frac{4L}{3};\frac{\sqrt{4}L}{\pi })$

Answer 1

The correct option is B $(\frac{L}{3};\frac{\sqrt{3}L}{\pi })$

As the Incident ray is horizontal 0 deg, and reflected ray is vertical 90 deg.
So the slope at point of reflection should be $\frac{90-0}{2}={45}^o$.
The derivative of equation of mirror surface, we get
$\frac{dy}{dx}=2cos\left(\frac{\pi x}{L}\right)=tan\left({45}^o\right)=1$
So, $x=\frac{L}{3}$.
and using equation of mirror surface,
$y=\frac{\sqrt{3}L}{\pi }$