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 $1,-1,1$, then find the DCs of the reflected ray.

• 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

The correct option is D $-\frac{1}{3},-\frac{2}{3},\frac{2}{3}$

Suppose the source of light be situated at $A(a,0,0)$

where $a\ne 0$, AO=incident ray, OB=reflected ray.

ON=normal to the mirror at 0.

$\therefore \angle AON=\angle NOB=\frac{\theta }{2}$ (say)

Direction ratios of normal place are $(1,-1,1)$

$\therefore$ Direction cosine of normal : $(\frac{1}{\sqrt{1^2+(-1{)}^2+1^2}},\frac{1}{\sqrt{1^2+(-1{)}^2+1^2}},\frac{1}{\sqrt{1^1(-1{)}^2+1^2}})$

$=(\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}})$

Hence, $\cos \left(\frac{\theta }{2}\right)=\frac{1}{\sqrt{3}}$

Direction ratio of $AO$ are $(\alpha ,0,0)$ and direction cosines are $(1,0,0)$

$\therefore$ Now let l,m,n be direction cosines of reflected ray $OB$.

$\therefore \frac{1+1}{2\cos \frac{\theta }{2}}=\frac{1}{\sqrt{3}}$

$\Rightarrow 1+1=\frac{2}{\sqrt{3}}\cos \frac{\theta }{2}=dfrac23$

$\Rightarrow 1=\frac{2}{3}-1=\frac{-1}{3}$

$=\frac{m+o}{2\cos \left(\frac{\theta }{2}\right)}=\frac{-1}{\sqrt{3}}$

$\Rightarrow m=\frac{-2}{\sqrt{3}}\cos \frac{\theta }{2}=\frac{-2}{3}$

$\frac{n+o}{2\cos \left(\frac{\theta }{2}\right)}=\frac{1}{\sqrt{3}}$

$\Rightarrow n=\frac{2}{\sqrt{3}}\cos \frac{\theta }{2}=\frac{2}{3}$

$\therefore$ Direction cosines of reflected ray are $(\frac{-1}{3},\frac{-2}{3},\frac{2}{3})$