Question

A ray of light passng through the point $(1,2)$ reflects on the $x$-axis at point $A$ and the reflected ray passes through the point $(5,3)$. Find the co-ordinates of $A$.

Answer 1

Let $B=(1,2)$ & $C=(5,3)$

$\because$ Ray $BA$ make angle $\theta$ with normal.

$\therefore$ $BA$ make angle ${(90+\theta )}^0$ with $x-$axis.

$AC$ makes $\measuredangle \theta$ with normal.

$\therefore$ $AC$ makes $(90-\theta )$ with $x-$axis.

Now, Slope of $AB=\tan {(90+\theta )}^0=\frac{2-\theta }{1-x}$

$\Rightarrow -\cot {\theta }^0=\frac{2}{1-x}$

Slope of $AC=\tan {(90-\theta )}^0=\frac{3-0}{5-x}$

$\Rightarrow \cot \theta =\frac{3}{5-x}$

$\Rightarrow \frac{2}{1-x}=\frac{-3}{5-x}\Rightarrow 10-2x=-3+3x$

$\Rightarrow 5x=13$

$\Rightarrow x=13/5$

$\therefore$ Co-ordinates of $A$ as $(13/5,0)$