Question

A ray of light coming from point $(1,2)$ is reflected by a point A on $x$-axis & is passed through the point $(5,3)$. The coordinates of A is $(\frac{13}{m},0)$.Find $m$

Answer 1

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

Let AL be the perpendicular to the x-axis.

By Law of reflection, we know,

angle of incidence = angle of reflection.

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

Let $\angle CAX=\theta$

$\angle OAB=180°-(\theta +2\varphi )$

$\Rightarrow =180°-(\theta +2(90°-\theta ))\{\because \varphi =90°-\theta \}$

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

$\Rightarrow \angle BAX=180°-\theta$

As we know that, slope of a line passing through $(x_1,y_1)$ and $(x_2,y_2)$ is-

$m=\frac{y_2-y_1}{x_2-x_1}$

And slope $m=\tan \theta$, where $\theta$ is the angle of inclination.

Slope of the line AC,

$m_{AC}=\frac{3-0}{5-x}$

also $m_{AC}=\tan \theta$

$\Rightarrow \tan \theta =\frac{3}{5-x}⟶\left(1\right)$

Slope of the line AB,

$m_{AB}=\frac{2-0}{1-x}$

$\therefore \tan (180°-\theta )=\frac{2}{1-x}$

As we know that,

$\tan (180°-\theta )=-\tan \theta$

$\therefore -\tan \theta =\frac{2}{1-x}$

$\tan \theta =\frac{2}{x-1}⟶\left(2\right)$

From ${eq}^n\left(1\right)\&\left(2\right)$, we have

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

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

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

$\Rightarrow 5x=13$

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

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

As given coordinates are $(\frac{13}{m},0)$.

$m=5$

Hence, the correct answer is 5.