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

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

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

( $\frac{1}{3}, \frac{2}{3}, \frac{2}{3}$ )

answer is D.

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Detailed Solution

Direction cosines of ON are $(\frac{1}{\sqrt{3}}, \frac{- 1}{\sqrt{3}}, \frac{1}{\sqrt{3}})$

Let $\frac{θ}{2} =$ angle between OA,ON= angle between OB,ON
$∴ \cos \frac{θ}{2} = \frac{1}{\sqrt{3}}$
Let( l,m,n) be the direction cosines of the reflected ray OB

$\frac{l + 1}{2 \cos \frac{θ}{2}} = \frac{1}{\sqrt{3}}, \frac{m + 0}{2 \cos \frac{θ}{2}} = - \frac{1}{\sqrt{3}}, \frac{n + 0}{2 \cos \frac{θ}{2}} = \frac{1}{\sqrt{3}} \Rightarrow l + 1 = \frac{2}{3}, m = \frac{- 2}{3}, n = \frac{2}{3} \Rightarrow l = \frac{- 1}{3}, m = \frac{- 2}{3}, n = \frac{2}{3}$