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Q.
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 $A$.
Straight Lines
Solution:
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
$tan\,\theta=\frac{3}{5-x}\quad\ldots\left(i\right)$
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 $tan\,\theta=\frac{2}{x-1} \quad\ldots\left(ii\right)$
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 $\left(\frac{13}{5}, 0\right)$.
