Question

If the straight line through the point $P(3,4)$ makes an angle $\frac{\pi }{6}$ with the $x$-axis and meets the line $3x+5y+1=0$ at $Q,$ then length of $PQ$ is

• A. $30$ units
• B. $30(\sqrt{3}-1)$ units
• C. $\sqrt{3}-1$ units
• D. $30(3\sqrt{3}-5)$ units

Answer 1

The correct option is D $30(3\sqrt{3}-5)$ units

Parametric equation of the line making an angle of $\frac{\pi }{6}$ with $x-$axis and passing through $(3,4)$ is
$x=3+r\cos \frac{\pi }{6};y=4+r\sin \frac{\pi }{6}\Rightarrow x=3+\frac{\sqrt{3}r}{2};y=4+\frac{r}{2}$
Substituting $x$ and $y,$ we get
$9+\frac{3\sqrt{3}r}{2}+20+\frac{5r}{2}+1=0$
$\Rightarrow r(3\sqrt{3}+5)=-60\Rightarrow r=-\frac{60}{3\sqrt{3}+5}=-\frac{60(3\sqrt{3}-5)}{27-25}=-30(3\sqrt{3}-5)$
Hence the length of $PQ$ is $r=30(3\sqrt{3}-5)$ units
($\because$ distance is positive quantity)