Question

A ray of light coming from the point $(1,2)$ is reflected at a point $A$ on the $x$-axis and then passes through the point $(5,3)$. The co-ordinates of the point $A$ is

• A. $(\frac{13}{5},0)$
• B. $(\frac{5}{13},0)$
• C. $(-7,0)$
• D. None of these

Answer 1

Answer: (A)

$(\frac{13}{5},0)$

Let the co-ordinates of $A$ be $(a,0)$.

Then the slope of the reflected ray is
$\frac{3-0}{5-a}=\tan \theta \left(\text{say}\right)$ ..... (1)
Then the slope of the incident ray $=\frac{2-0}{1-a}=\tan (\pi -\theta )$ ..... (2)

Since $\tan \theta +\tan (\pi -\theta )=0$
From (1) and (2),

$\Rightarrow \frac{3}{5-a}+\frac{2}{1-a}=0$

$\Rightarrow 3-3a+10-2a=0$

$\Rightarrow a=\frac{13}{5}$
Thus, the co-ordinates of $A$ are $(\frac{13}{5},0)$.