Question

Two plane mirrors are inclined at angle $\theta$ as shown in figure. If a ray parallel to $OB$ strikes the other mirror at $P$ and finally emerges parallel to $OA$ after two reflections, then $\theta$ is equal to

• A. ${90}^{\circ }$
• B. ${60}^{\circ }$
• C. ${45}^{\circ }$
• D. ${30}^{\circ }$

Answer 1

The correct option is B ${60}^{\circ }$


$N_1$ and $N_2$ are the normal to mirrors $M_1$ and $M_2$ respectively.
From geometry, the incident angle at point $P$ will be
$\angle i=\frac{\pi }{2}-\theta =\angle r$
From $\Delta OPC$,
the incident angle at point $C$ will be
$\angle i=\frac{\pi }{2}-[\pi -(\theta +\theta \left)\right]=2\theta -\frac{\pi }{2}$
$\Rightarrow \angle r=\angle i=2\theta -\frac{\pi }{2}$..........(1)
Since the reflected ray is parallel to $M_1$ or $OA$ line.
So, from geometry, the reflected angle at point $C$ will be
$\angle r=\frac{\pi }{2}-\theta$ ..........(2)
Equating both equations (1) and (2)
$2\theta -\frac{\pi }{2}=\frac{\pi }{2}-\theta$
$\Rightarrow \pi =3\theta$
$\Rightarrow \theta =\frac{\pi }{3}={60}^{\circ }$
Hence, option $\left(b\right)$ is the correct answer.