If the straight line passing through $P(3,4)$ makes an angle $\frac{\pi}{6}$ with the positive $x$-axis in…
If the straight line passing through $P(3,4)$ makes an angle $\frac{\pi}{6}$ with the positive $x$-axis in anticlockwise direction and meets the line $12 x+5 y+10=0$ at Q , then the length of the segment PQ is
$\frac{64}{12 \sqrt{2} 1}$
$\frac{96}{9 \sqrt{2}-1}$
$\frac{112}{10 \sqrt{3}+3}$
$\frac{132}{12 \sqrt{3}+5}$
Solution
Any line through $P(3,4)$ making an angle with $x$-axis
is $\frac{x-3}{\cos 30^{\circ}}=\frac{y-4}{\sin 30^{\circ}}=r$
Any point $Q$ on it is $\left(\frac{r \sqrt{3}}{2}+3, \frac{r}{2}+4\right)$
and $Q$ lies on $12 x+5 y+10=0$
$\begin{aligned}
& \therefore 12\left[\left(\frac{r \sqrt{3}}{2}\right)+3\right]+5\left(\frac{r}{2}+4\right)+10=0 \\
& \Rightarrow r=\frac{132}{12 \sqrt{3}+5}=\text { length of segment } P Q
\end{aligned}$