Question

A ray of light is sent along the line which passes through the point $(2,3)$. The ray is reflected from the point $P$ on $x-$axis. If the reflected ray passes through the point $(6,4)$ then the coordinates of $P$ are

• A. $(\frac{26}{7},0)$
• B. $(0,\frac{26}{7})$
• C. $(-\frac{26}{7},0)$
• D. None of these

Answer 1

The correct option is A $(\frac{26}{7},0)$

Let $P\equiv (\alpha ,0)$

Let the reflected ray makes an angle $\theta$ with +ve direction of x-axis, then the incident ray makes angle $(\pi -\theta )$ with positive direction of x-axis.

The slope of the incident ray is

$=\frac{0-3}{\alpha -2}=\tan (\pi -\theta )\Rightarrow \tan \theta =\frac{3}{\alpha -2}$ ...(1)

The slope of the reflected ray is

$=\frac{4-0}{6-\alpha }=\tan \theta \Rightarrow \tan \theta =\frac{4}{6-\alpha }$ ...(2)

From (1) and (2), we get

$\frac{3}{\alpha -2}=\frac{4}{6-\alpha }\Rightarrow 18-3\alpha =4\alpha -8\Rightarrow 7\alpha =26\Rightarrow \alpha =\frac{26}{7}$

$\therefore$ The coordinate of $P$ are $(\frac{26}{7},0)$