Two plane mirrors are inclined to each other at angle . A ray of light is reflected first at one mirror and then a

Two plane mirrors are inclined to each other at angle $\theta$ . A ray of light is reflected first at one mirror and then at the other. Find the total deviation of the ray
Option: 1
$360^{\circ}-2 \theta$

Option: 2
$360^{\circ}+2 \theta$

Option: 3
$180^{\circ}-2 \theta$

Option: 4
$180^{\circ}+2 \theta$

Answers (1)

A ray $\mathrm{A B}$ is incident on mirror $\mathrm{O M_{1}}$ at angle $\mathrm{\alpha}$ and is reflected along $\mathrm{B C}$ suffering a deviation
$\mathrm{\delta_{1}=\angle \mathrm{FBC}}$

The ray $\mathrm{\mathrm{BC}}$ falls on mirror $\mathrm{\mathrm{OM}_{2}}$ at an angle of incidence $\mathrm{\beta }$ and is reflected along $\mathrm{CD }$ suffering another deviation $\mathrm{\delta_{2}=\angle \mathrm{GCD} }$
The total deviation is $\mathrm{\delta=\delta_{1}+\delta_{2} }$

It is clear from the diagram that
$\delta_{1}=180^{\circ}-2 \alpha$ and $\delta_{2}=180^{\circ}-2 \beta$
$\therefore \quad \delta=\delta_{1}+\delta_{2}=360^{\circ}-2(\alpha+\beta)$

Now, in triangle $\mathrm{OBC}, \angle \mathrm{OBC}+\angle \mathrm{BCO}+\angle \mathrm{BOC}=180^{\circ}$

$\mathrm{or \quad\left(90^{\circ}-\alpha\right)+\left(90^{\circ}-\beta\right)+\theta=180^{\circ}}$
$\mathrm{or \quad \alpha+\beta=\theta}$
$\mathrm{Hence \: \delta=360^{\circ}-2 \theta}$

Which is independent of the angle of incidence $\mathrm{\alpha}$ at the first mirror.