A ray of light coming from the point

Question:

A ray of light coming from the point $(2,2 \sqrt{3})$

is incident at an angle $30^{\circ}$ 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 $\mathrm{AB}$ passes through the point:

  1. $\left(3,-\frac{1}{\sqrt{3}}\right)$

  2. $(3,-\sqrt{3})$

  3. $\left(4,-\frac{\sqrt{3}}{2}\right)$

  4. $(4,-\sqrt{3})$


Correct Option: , 2

Solution:

For point $\mathrm{A}$

$\tan 60^{\circ}=\frac{2 \sqrt{3}-\mathrm{k}}{2-1}$

$\sqrt{3}=2 \sqrt{3}-\mathrm{k}$

$\therefore \mathrm{k}=\sqrt{3}$

so point $\mathrm{A}(1, \sqrt{3})$

Now slope of line $\mathrm{AB}$ is $\mathrm{m}_{\mathrm{AB}}=\tan 120^{\circ}$

$\mathrm{m} \mathrm{m}_{\mathrm{AB}}=-\sqrt{3}$

Now equation of line $A B$ is

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

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

Now satisfy options

 

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