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 $12x+5y+10=0$ at $Q$, find the length of $PQ$.

Answer 1

Any line through $P(3,4)$ making an angle $\frac{\pi }{6}$ with $x-$axis is
$\frac{x-3}{\cos {30}^o}=\frac{y-4}{\sin {30}^o}=r$
Where $r$ represents the distance of any point on this line from the given point $P(3,4)$
Any point $Q$ on it is $\left(r\sqrt{3}/2\right)+3,\left(r/2\right)+4$ and $Q$ lies on $12x+5y+10=0$
$\therefore 12[\left(\frac{r\surd 3}{2}\right)+3]+5(\frac{r}{2}+4)+10=0$
$\therefore (12\sqrt{3}+5)r+132=0$
$\therefore r=\frac{|-132|}{12\surd \left(3\right)+5}=\frac{132}{12\surd \left(3\right)+5}.$