A circle 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 3x+y−6=0 and the point D is (332, 32). Further, it is given that the origin and the centre of C are on the same side of the line PQ. The equation of circle C is

Slope of CD=13 ∴ Parametric equation of CD is x−33232=y−3212=±1∴ Two possible coordinates of C are (32+332,12+32) or (−32+332,−12+32) i.e., (23,2) or (3,1)As (0, 0) and C lie on the same side of PQ∴(3,1) should be the coordinates of C. Equation of the circle is (x−3)2+(y−1)2=1

A circle 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 3x+y−6=0 and the point D is (332, 32). Further, it is given that the origin and the centre of C are on the same side of the line PQ. The equation of circle C is

Slope of CD=13 ∴ Parametric equation of CD is x−33232=y−3212=±1∴ Two possible coordinates of C are (32+332,12+32) or (−32+332,−12+32) i.e., (23,2) or (3,1)As (0, 0) and C lie on the same side of PQ∴(3,1) should be the coordinates of C. Equation of the circle is (x−3)2+(y−1)2=1

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A circle 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 is $(\frac{3 \sqrt{3}}{2}, \frac{3}{2})$ . Further, it is given that the origin and the centre of C are on the same side of the line PQ. The equation of circle C is

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

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

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

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

answer is D.

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Detailed Solution

Slope of $C D = \frac{1}{\sqrt{3}}$
$∴$ Parametric equation of CD is $\frac{x - \frac{3 \sqrt{3}}{2}}{\frac{\sqrt{3}}{2}} = \frac{y - \frac{3}{2}}{\frac{1}{2}} = \pm 1$
$∴$ Two possible coordinates of C are
$(\frac{\sqrt{3}}{2} + \frac{3 \sqrt{3}}{2}, \frac{1}{2} + \frac{3}{2}) o r (\frac{- \sqrt{3}}{2} + \frac{3 \sqrt{3}}{2}, - \frac{1}{2} + \frac{3}{2})$
$i . e ., (2 \sqrt{3}, 2) o r (\sqrt{3}, 1)$
As (0, 0) and C lie on the same side of PQ
$∴ (\sqrt{3}, 1)$ should be the coordinates of C.
Equation of the circle is ${(x - \sqrt{3})}^{2} + {(y - 1)}^{2} = 1$