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Basics of a straight line - class-X
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If the straight line through the point $P(3,4)$ makes an angle $\cfrac{\pi}{6}$ with the x-axis and meets the line $3x+5y+1=0$ at $Q$, the length $PQ$ is
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A
$\dfrac {132}{5\sqrt {3}-12}$
💡 Explanation:
The equation of straight line passing through $P(3,4)$ is $y=\tan \dfrac{\pi}{6}{x}+(4-3\tan \dfrac{\pi}{6})\implies y=\dfrac{x}{\sqrt{3}}+4-\sqrt{3}$
The point of intersection will be $\bigg(\dfrac{55-57\sqrt{3}}{5+3\sqrt{3}},\dfrac{-10+3\sqrt{3}}{5+3\sqrt{3}}\bigg)$
Length will be $30(5-3\sqrt{3})$