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. $\left(-7,0\right)$
• D. None of these

Answer 1

The correct option is A

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

Explanation for the correct option:

Find the coordinates of point $A$:

Let the incident ray touches the x-axis at $A(x,0)$.

From figure,

slope of $PA$, $\begin{array}{rcl}{m}_1& =& \tan \theta \end{array}$

$\begin{array}{rcl}& =& \frac{2}{(1–x)}\end{array}$

Slope of $QA$, $m_2=\tan (\pi –\theta )$

$\Rightarrow$ $–\tan \theta =\frac{3}{(5–x)}$

$\therefore m_1=-m_2$

$\Rightarrow$ $\frac{2}{(1-x)}=\frac{-3}{(5–x)}$

$\Rightarrow$$10–2x=-3+3x$

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

Thus, the co-ordinates of point $A$ are $\left(\frac{13}{5},0\right)$

Hence, Option ‘A’ is Correct.