Question

A ray of light coming from the point \((2,\,2\sqrt{3})\) is incident at an angle \(30^\circ\) on the line \(x=1\) at the point A. The ray gets reflected on the line \(x=1\) and meets x-axis at the point B. Then, the line AB passes through the point :

• A. \((3,\,-\sqrt{3})\)
• B. \((4,\,-\sqrt{3})\)
• C. \(\left(4,\,-\frac{\sqrt{3}}{2}\right)\)
• D. \(\left(3,\,-\frac{1}{\sqrt{3}}\right)\)

Hint:

Take the 30 degrees as the angle between the ray and the line x = 1 (so the ray has slope of magnitude \(\sqrt{3}\)); find A, then the reflected ray's slope is reversed.

Answer 1

The Correct Option is A

Let the source be \(P(2,2\sqrt{3})\). The ray makes \(30^\circ\) with the vertical line \(x=1\), so the incident ray has angle \(60^\circ\) with the x-axis, i.e. slope of magnitude \(\tan 60^\circ=\sqrt{3}\).
It travels horizontally from \(x=2\) to \(x=1\) (a run of 1), so it drops by \(\sqrt{3}\): \(A=(1,\,2\sqrt{3}-\sqrt{3})=(1,\sqrt{3})\).
Reflection in the vertical line \(x=1\) reverses the horizontal direction only, so the reflected ray has slope \(-\sqrt{3}\) (going right and downward).
Line AB: \(y-\sqrt{3}=-\sqrt{3}(x-1)\), i.e. \(y=-\sqrt{3}(x-2)\). It meets the x-axis at \(B=(2,0)\).
Check the options: at \(x=3\), \(y=-\sqrt{3}\). So the line passes through \((3,-\sqrt{3})\); at \(x=4\), \(y=-2\sqrt{3}\) which is not in the options.
Answer: (1) \((3,\,-\sqrt{3})\)