Let $P$ is a point on the line $y+2x=2$ and $Q$ and $R$ are two points on the line $3y+6x=3$. If the triangle $PQR$ is an equilateral triangle, then its area (in sq. units) is equal to

• A

  $\frac{1}{15}$
• Correct answer
  B

  $\frac{1}{5\sqrt{3}}$
• C

  $\frac{1}{3\sqrt{5}}$
• D

  $\frac{1}{2\sqrt{15}}$

Solution

The distance between the given parallel lines $\left(\mathrm{h}\right)$ is $\frac{2-1}{\sqrt{5}}=\frac{1}{\sqrt{5}}$

$\therefore$ Length of the side of the triangle is $=\frac{2h}{\sqrt{3}}$
Hence, the area of triangle $=\frac{\sqrt{3}}{4}\times \frac{4h^2}{3}=\frac{h^2}{\sqrt{3}}=\frac{1}{5\sqrt{3}}$ sq. units

Hence, the correct answer is option: 'B'.