Question

A light ray emits from the origin making an angle $30^{\circ}$ with the positive $x$-axis. After getting reflected by the line $x+y=1$, if this ray intersects $x$-axis at $Q$, then the abscissa of $Q$ is

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

Answer 1

Answer: (C)

Slope of reflected ray $=\tan 60^{\circ}=\sqrt{3}$
Line $y=\frac{x}{\sqrt{3}}$ intersect $y+x=1$ at $\left(\frac{\sqrt{3}}{\sqrt{3}+1}, \frac{1}{\sqrt{3}+1}\right)$
Equation of reflected ray is
$y-\frac{1}{\sqrt{3}+1}=\sqrt{3}\left(x-\frac{\sqrt{3}}{\sqrt{3}+1}\right)$
Put $y =0 \Rightarrow x =\frac{2}{3+\sqrt{3}}$