Question

Two plane mirrors are inclined at an angle $\theta$ as shown in the figure. Find the value of $\theta$ such that the light ray retraces its path after second reflection.

• A. ${30}^{\circ }$
• B. ${60}^{\circ }$
• C. ${45}^{\circ }$
• D. ${75}^{\circ }$

Answer 1

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

The light ray will retrace its path after the second reflection if it strikes the horizontal plane mirror normally at the point of incidence $\left(M\right)$.


From the geometry of the ray diagram,
Angle of incidence at the inclined mirror at point of incidence $\left(N\right)$,
$i={90}^{\circ }-{30}^{\circ }={60}^{\circ }$
So, $i=r={60}^{\circ }$
(from the laws of reflection)
In $△ONM$,
$\angle ONM={90}^{\circ }-r={90}^{\circ }-{60}^{\circ }={30}^{\circ }$
Now, again in $△ONM$,
$\angle NOM+\angle ONM+\angle NMO={180}^{\circ }$
$\Rightarrow \theta +{30}^{\circ }+{90}^{\circ }={180}^{\circ }$
$\Rightarrow \theta ={180}^{\circ }-{120}^{\circ }={60}^{\circ }$