Question

Distance between two parallel lines is unity. A point $P$ lies between the lines at a distance ${}^{\prime }{a}^{\prime }$ from one of them. Find the length of a side of an equilateral $△PQR$ such that $Q$ lies on one of the parallel lines while $R$ lies on the other.

• A. $\frac{2}{\sqrt{3}}\sqrt{a^3-a^2+1}$
• B. $\frac{2}{\sqrt{3}}\sqrt{a^2-a}+1$
• C. $\frac{2}{\sqrt{3}}\sqrt{a^2-a+1}$
• D. $\frac{2}{\sqrt{3}}\sqrt{a^3-a^2}+1$

Answer 1

The correct option is C $\frac{2}{\sqrt{3}}\sqrt{a^2-a+1}$

Let x be the length of the side of the equilateral triangle PQR (see fig.).
Then, we have $x\sin \theta =a\cdots \left(1\right)$
and $x\sin (\frac{\pi }{3}-\theta )=1-a\cdots \left(2\right)$
elliminating $\theta$, we have
$\frac{\sqrt{3}}{2}\sqrt{x^2-a^2}-\frac{a}{2}=1-a$
i.e., $x=\frac{2}{\sqrt{3}}\sqrt{a^2-a+1}$