Question

$P$ is a point on the line $y+2x=1$ and $Q$, and $R$ are two points on the line $3y+6x=6$ such that triangle $PQR$ is an equilateral triangle. The length of the side of the triangle is

• A. $\frac{2}{\sqrt{15}}$
• B. $\frac{3}{\sqrt{15}}$
• C. $\frac{4}{\sqrt{15}}$
• D. $\frac{5}{\sqrt{15}}$

Answer 1

The correct option is A

$\frac{2}{\sqrt{15}}$

$y+2x=1\dots \left(1\right)$

$3y+6x=6\Rightarrow y+2x=2\dots \left(2\right)$

$PM$ is the distance between the lines (1) & (2) $=\frac{1}{\sqrt{2^2+1}}=\frac{1}{\sqrt{5}}$

$\because △PQR$ is equilateral, every angle is equal to ${60}^{\circ }$

$\Rightarrow \sin {60}^{\circ }=\frac{PM}{PQ}$

$\Rightarrow PQ=\frac{2}{\sqrt{15}}$