Question

A ray of light coming from the point $(2,2\surd 3)$ is incident at an angle $30°$ on the line $x=1$ at the point $A$. The ray gets reflected on the line $x=1$ and meets x-axis at the point $B$. Then, the line $AB$ passes through the point:

• A. $(4,-\surd 3)$
• B. $\left(3,\frac{-1}{\surd 3}\right)$
• C. $(3,-\surd 3)$
• D. $\left(4,\frac{-\surd 3}{2}\right)$

Answer 1

The correct option is C

$(3,-\surd 3)$

Explanation for the correct option:

Step 1. Find the equation of line $AB$:

From the figure, we have

For point $A$

$\begin{array}{rcl}\tan 60°& =& \frac{2\sqrt{3}-k}{2-1}\end{array}$

$\Rightarrow$ $\begin{array}{rcl}\sqrt{3}& =& 2\sqrt{3}-k\end{array}$

$\Rightarrow$ $\begin{array}{rcl}k& =& \sqrt{3}\end{array}$

So, point $A$ is $\left(1,\sqrt{3}\right)$

Step 2. slop of line $AB$ is $m_{AB}=\tan 120°$

$\Rightarrow$ $m_{AB}=-\sqrt{3}$

Now, equation of line $AB$ is

$y-\sqrt{3}=-\sqrt{3}\left(x-1\right)$

$\Rightarrow$$\sqrt{3}x+y=2\sqrt{3}$

Now satisfy the given options, we get

$\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{-}\mathbf{\surd }\mathbf{3}\mathbf{)}$ satisfy the line.

Hence, Option ‘C’ is Correct.