Question

If the straight line drawn through the point $P(\sqrt{3},2)$ and inclined at an angle of $\frac{\pi }{6}$ with the positive direction of $x-$axis meets the line $\sqrt{3}x-4y+8=0$ at point $Q,$ then the length of $PQ$ is

Answer 1

Using parametric form of straight line $Q\equiv (\sqrt{3}+\frac{\sqrt{3}r}{2},2+\frac{r}{2})$
Here, $r=PQ$

$Q$ lies on $\sqrt{3}x-4y+8=0$
$\Rightarrow \sqrt{3}(\sqrt{3}+\frac{\sqrt{3}r}{2})-4(2+\frac{r}{2})+8=0$
$\Rightarrow 3+\frac{3}{2}r-8-2r+8=0$
$\Rightarrow \frac{r}{2}=3$
$\Rightarrow r=6$
$\therefore PQ=6$