Question

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 $(5,3)$. The coordinates of the point $A$ are

• A. $\left(\frac{13}{5},0\right)$
• B. $\left(\frac{5}{13},0\right)$
• C. $(-7,0)$
• D. None of these

Answer 1

Answer: (A)

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

Let the coordinates of point A be $(a,0)$
Let OL be the normal .
$\Rightarrow \angle BAL=\angle CAL=\varphi$
Let $\angle CAX=\theta$
$\Rightarrow \angle OAB=\theta$
So, $\angle BAX={90}^0+\varphi$
Slope of line Ac $=\tan ({90}^0-\varphi )=\frac{3}{5-a}$
$\Rightarrow \cot \varphi =\frac{3}{5-a}$ ......(i)

Slope of line AB $=\tan ({90}^0+\varphi )=\frac{2}{1-a}$
$\Rightarrow \cot \varphi =\frac{2}{a-1}$ ......(ii)
From (i) and (ii),

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

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

Hence, the image is $(\frac{13}{5},0)$