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 co-ordinates of the point $A$ is

• 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)

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

Answer 2

Let the coordinates of point $A$ be $( a , 0)$ Let $OL$ be the normal. $\Rightarrow \angle BAL =\angle CAL =\phi$ Let $\angle CAX =\theta$ $\Rightarrow \angle OAB =\theta$ So, $\angle BAX =90^{\circ}+\phi$ Slope of line $A c=\tan \left(90^{\circ}-\phi\right)=\frac{3}{5-a}$ $\Rightarrow \cot \phi=\frac{3}{5- a }$ ....(i) Slope of line $AB =\tan \left(90^{\circ}+\phi\right)=\frac{2}{1- a }$ $\Rightarrow \cot \phi=\frac{2}{a-1}$ .... (ii) From (i) and (ii), $\Rightarrow \frac{3}{5- a }=\frac{2}{ a -1}$ $\Rightarrow a =\frac{13}{5}$ Hence, the image is $\left(\frac{13}{5}, 0\right)$