Question

If $P(0,0),Q(1,0)$ and $R(\frac{1}{2},\frac{\sqrt{3}}{2})$ are three given points, then the centre of the circle for which the lines $PQ,QR$ and $RP$ are the tangents is

• A. $(\frac{1}{2},\frac{1}{4})$
• B. $(\frac{1}{2},\frac{\sqrt{3}}{4})$
• C. $(\frac{1}{2},\frac{1}{2\sqrt{3}})$
• D. $(\frac{1}{2},\frac{-1}{\sqrt{3}})$

Answer 1

The correct option is C $(\frac{1}{2},\frac{1}{2\sqrt{3}})$

$P(0,0),Q(1,0)$ and $R(\frac{1}{2},\frac{\sqrt{3}}{2})$
$\overline{PQ}=\overline{QR}=\overline{RP}=1$
$\therefore △PQR$ is equilateral triangle.

Hence, centre of circle is the centroid of $△PQR.$
$\Rightarrow$ Centre of circle : $(\frac{1}{2},\frac{1}{2\sqrt{3}})$