question_answer

A mirror and a source of light are situated at the origin 0 and at a point on OX respectively. A ray of light from the source strikes the mirror and is reflected. If the direction ratios of the normal to the plane are 1, -1, 1, then direction consines of the reflected rays are

A) \[\frac{1}{3},\frac{2}{3},\frac{2}{3}\]

B) \[-\frac{1}{3},\frac{2}{3},\frac{2}{3}\]

C) \[-\frac{1}{3},\frac{2}{3},-\frac{2}{3}\]

D) \[-\frac{1}{3},-\frac{2}{3},\frac{2}{3}\]

Correct Answer: D

Solution :

[d] Let the ray of light comes along x-axis and strikes the mirror at the origin. Direction cosines of normal are \[\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}so.\cos \frac{\theta }{2}=\frac{1}{\sqrt{3}}\] Let the reflected ray has direction cosines l, m, n then \[\frac{l+1}{2\cos \frac{\theta }{2}}=\frac{1}{\sqrt{3}}\Rightarrow l=\frac{2}{3}-1=-\frac{1}{3}\] \[\frac{m+0}{2\cos \frac{\theta }{2}}=-\frac{1}{\sqrt{3}}\Rightarrow m=-\frac{2}{3}\] \[\frac{n+0}{2\cos \frac{\theta }{2}}=\frac{1}{\sqrt{3}}\Rightarrow n=\frac{2}{3}\]