Straight Lines
Reflection of a ray and intersection with axis
nta_pyq_2023_jan
Grade 11
Question:
A light ray emits from the origin making an angle $30°$ 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
$\dfrac{2}{(\sqrt{3}-1)}$
$\dfrac{2}{3+\sqrt{3}}$
$\dfrac{2}{3-\sqrt{3}}$
$\dfrac{\sqrt{3}}{2(\sqrt{3}+1)}$
Step-by-Step Solution
Key Concept: Find the point of intersection of the incident ray $y = \frac{x}{\sqrt{3}}$ with $x + y = 1$. The reflected ray has slope $\tan 60° = \sqrt{3}$ (by reflection rule about $x+y=1$). Write the reflected ray's equation and set $y = 0$.
Incident ray meets $x+y=1$ at $\left(\frac{\sqrt{3}}{\sqrt{3}+1}, \frac{1}{\sqrt{3}+1}\right)$. Reflected ray slope $= \sqrt{3}$. Setting $y=0$: $x = \frac{2}{3+\sqrt{3}}$.
Correct Answer: 2