Question

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

• A. $(\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}})$
• B. $(-\frac{1}{3},-\frac{2}{3},\frac{2}{3})$
• C. $(\frac{2}{3},-\frac{1}{3},-\frac{2}{3})$
• D. $(-\frac{2}{3},-\frac{2}{3},\frac{1}{3})$

Answer 1

The correct option is B $(-\frac{1}{3},-\frac{2}{3},\frac{2}{3})$

Let the source of light be situated at $A(a,0,0),$ where $a\ne 0$
and $OA$ is incident ray, $OB$ is reflected ray.
$ON$ is the normal to the mirror at $O.$
$\Rightarrow \angle AON=\angle NOB=\frac{\theta }{2}$(let)

$D.r^{\prime }s$ of $OA=(a,0,0)$
$\Rightarrow D.c^{\prime }s$ are $(1,0,0)$
and $D.c^{\prime }s$ of $ON$ are $(\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$
$\Rightarrow \cos \frac{\theta }{2}=l_1{l}_2+m_1{m}_2+n_1{n}_2\Rightarrow \cos \frac{\theta }{2}=\frac{1}{\sqrt{3}}\cdots \left(i\right)$
Let $l,m,n$ are the direction cosines of reflected ray $OB,$ 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 $\frac{n+0}{2\cos \frac{\theta }{2}}=\frac{1}{\sqrt{3}}$
using $\cos \frac{\theta }{2}=\frac{1}{\sqrt{3}}$ in the above equations,
on solving we get,
$l=-\frac{1}{3},m=-\frac{2}{3},n=\frac{2}{3}$