Question

A mirror and a source of light are situated at the origin O 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 proportional to $1$, $-$1, $1$, then direction cosines of the reflected ray 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. $-\frac{1}{3},-\frac{2}{3},\frac{2}{3}$

Answer 1

Answer: (D)

$-\frac{1}{3},-\frac{2}{3},\frac{2}{3}$

Let the source of light be situated at $A(a,0,0)$ where $a\ne 0$
Let $OA$ be the incident ray and $OB$ the reflected ray.
$ON$ is the normal to the mirror at $O$
$\therefore \angle AON=\angle NOB=\frac{\theta }{2}$ (let)
D.R's of $OA$ are $0,0,1$
and so its D.R's are $1,0,0$
D.R's of $ON$ are $\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}}$
$\therefore \cos \frac{\theta }{2}=\frac{1}{\sqrt{3}}$
Let $l,m,n$ be the d.R's of the reflected ray $OA$. then
$\frac{l+1}{2\cos \frac{\theta }{2}}=\frac{1}{\sqrt{3}},\frac{m+0}{2\cos \frac{\theta }{2}}=-\frac{1}{\sqrt{3}}$ and $\frac{n+0}{2\cos \frac{\theta }{2}}=\frac{1}{\sqrt{3}}$
$\Rightarrow l=\frac{2}{3}-1,m=-\frac{2}{3},n=\frac{2}{3}$
$\Rightarrow l=\frac{-1}{3},m=\frac{-2}{3},n=\frac{2}{3}$
Hence, D.C's of the reflected ray are $\frac{-1}{3},\frac{-2}{3},\frac{2}{3}$