Mathematics

Problem

A circle

C

of radius 1 is inscribed in an equilateral triangle

PQR

. The points of contact of

C

with the sides

PQ

QR

RP

are

D

E

F

, respectively. The line

PQ

is given by the equation

\sqrt{3}x + y - 6 = 0

, and the point

D

\left(\frac{3\sqrt{3}}{2}, \frac{3}{2}\right)

. It is given that the origin and the centre of

C

are on the same side of the line

PQ

. Find the equation of circle

C

. a.

(x - 2\sqrt{3})^2 + (y - 1)^2 = 1

(x - 2\sqrt{3})^2 + \left(y + \frac{1}{2}\right)^2 = 1

(x - \sqrt{3})^2 + (y + 1)^2 = 1

(x - \sqrt{3})^2 + (y - 1)^2 = 1