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 horizontally becomes vertical after reflection. The coordinates of the point(s) where this ray is incident is

• A. True
• B. False

Answer 1

The correct option is B False

as the horizontal ray becomes vertical on striking the tangent at the point on the curve is at an angle ${135}^{\circ }$ with the horizontal
$y=\frac{2L}{\pi }sin\left(\frac{\pi x}{L}\right)$
$\frac{\partial y}{\partial x}=\frac{\partial \frac{2L}{\pi }sin\left(\frac{\pi x}{L}\right)}{\partial x}$
$tan\left({135}^{\circ }\right)=2cos\left(\frac{\pi x}{L}\right)$
$cos(\frac{\pi x}{L}=-\frac{1}{2}$
$\frac{\pi x}{L}=\frac{2\pi }{3}$
$x=\frac{2L}{3}$
$y=\frac{2L}{\pi }sin\left(\frac{\pi x}{L}\right)$
$y=\frac{2L}{\pi }sin\left(\frac{\pi \frac{2L}{3}}{L}\right)$
$y=\frac{\sqrt{3}\pi }{L}$
option $D$ is correct