Question

A ray of light passing through the point $A$ and the reflected ray passes 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 coordinates of $A$ .

Answer 1

Let the coordinates of point A be $(a,0)$.

Draw a line $\left(AL\right)$ perpendicular to the x-axis

We know that angle of incidence is equal to angle of reflection. Hence, let

$\angle BAL=\angle CAL=\varphi$

Let $\angle CAX=\theta$

$\therefore \angle OAB={180}^o-(\theta +2\varphi )={180}^o-[\theta +2({90}^o-\theta \left)\right]$

$={180}^o-\theta -{180}^o+2\theta$

$=\theta$

$\therefore \angle BAX={180}^o-\theta$

Now , slope of line $AC=\frac{3-0}{5-a}$

$\Rightarrow tan\theta =\frac{3}{5-a}...\left(i\right)$

Slope of line $AB=\frac{2-0}{1-a}$

$\Rightarrow tan({180}^o-\theta )=\frac{2}{1-a}$

$\Rightarrow tan\theta =\frac{2}{1-a}$

$\Rightarrow tan\theta =\frac{2}{a-1}...\left(2\right)$

From equations (1) and (2) we obtain

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

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

$\Rightarrow a=\frac{13}{5}$

Thus, the coordinates pf point A are

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