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 $(5,3)$. Find the coordinates of the point $A$.

• A. $(5,0)$
• B. $\left(\frac{13}{5},0\right)$
• C. $\left(\frac{-13}{5},0\right)$
• D. $\left(\frac{6}{5},0\right)$

Answer 1

Answer: (B)

Let the incident ray strike on $x$-axis at the point $A$ whose coordinates be $(x,0)$. From the figure, the slope of the reflected ray is given by
$tan\theta =\frac{3}{5-x}\dots \left(i\right)$



Again, the slope of the incident ray is given by
$tan\left(\pi -\theta \right)=\frac{-2}{x-1}$
or $-tan\theta =\frac{-2}{x-1}$ or $tan\theta =\frac{2}{x-1}\dots \left(ii\right)$
Solving $\left(i\right)$ and $\left(ii\right)$, we get
$\frac{3}{5-x}=\frac{2}{x-1}$ or $x=\frac{13}{5}$
Therefore, the required coordinates of the point $A$ are $\left(\frac{13}{5},0\right)$.