Question

The distance between two parallel lines is unity. A point P lies between the lines at a distance a from one of them. The length of a side of an equilateral $\Delta$PQR, vertex Q of which lies on one of the parallel lines and vertex R lies on the other line, is

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

$LetPQ=QR=RP=rAnd\angle PQX=\theta ,the\angle PQX=\frac{\pi }{3}+\theta$

Given, PL = a, RN = 1
Now, a = Pl = r sin θ and 1 = RN = $rsin[\frac{\pi }{3}+\theta ]$
$\Rightarrow 1=r[sin\frac{\pi }{3}cos\theta +\frac{1}{2}sin\theta ]\Rightarrow r[\frac{\sqrt{3}}{2}cos\theta +\frac{1}{2}sin\theta ]=1\Rightarrow r\frac{\sqrt{3}}{2}\sqrt{1-\frac{a^2}{r^2}+\frac{1}{2}.\frac{a}{r}}=1[\because sin\theta =\frac{a}{r}]\Rightarrow \frac{\sqrt{3}}{2}\sqrt{r^2-a^2}=1-\frac{a}{2}\Rightarrow r^2-a^2=\frac{(2-a{)}^2}{3}=\frac{4+a^2-4a}{3}\therefore r=\frac{2}{\sqrt{3}}\sqrt{a^2-a+1}$
Hence, (b) is the correct answer.