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 $Q$, the length $PQ$ is

• A. $\frac{132}{12\sqrt{3}+5}$
• B. $\frac{132}{12\sqrt{3}-5}$
• C. $\frac{132}{5\sqrt{3}+12}$
• D. $\frac{132}{5\sqrt{3}-12}$

Answer 1

Answer: (D)

$\frac{132}{5\sqrt{3}-12}$

The equation of straight line passing through $P(3,4)$ is $y=\tan \frac{\pi }{6}x+(4-3\tan \frac{\pi }{6})⟹y=\frac{x}{\sqrt{3}}+4-\sqrt{3}$

The point of intersection will be $(\frac{55-57\sqrt{3}}{5+3\sqrt{3}},\frac{-10+3\sqrt{3}}{5+3\sqrt{3}})$

Length will be $30(5-3\sqrt{3})$