Question

A ray of light passing through the point $(1,2)$ reflects on the axis of $x$ at the point $A$ and the reflected ray passes through the points $(5,3)$. Determine the coordinates of the point $A$.

Answer 1

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

Let AL be the perpendicular to x - axis.

By Law of reflection we know,

angle of incidence = angle of reflection.

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

Let $\angle CAX=\theta$

$\angle OAB=180\circ -(\theta +2\varphi )$

$=180\circ -[\theta +2(90\circ -\theta )$

$\because (\varphi =90-\theta )$

$\therefore \angle OAB=180\circ -\theta -180\circ +2\theta$

$=\theta \Rightarrow \angle BAX=180\circ -\theta$

Slope of the line AC,$m=3-05-a$

also $m=\tan \theta$

$\Rightarrow \tan \theta =35-a$-------(1)

Slope of the line $AB=2-01-a$

$\therefore \tan (180-\theta )=21-a$

But $\tan (180-\theta )=-\tan \theta$

$\therefore -\tan \theta =21-a$

$\therefore \tan \theta =2a-1$------(2)

equally (1) and (2) we get,

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

$\Rightarrow 3(a-1)=2(5-a)$

$3a-3=10-2a$

$\Rightarrow 5a=13$

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

Hence the coordinate of A are $(\frac{13}{5},0)$