question_answer

The circle \[{{C}_{1}}:{{x}^{2}}+{{y}^{2}}=3,\] with center at O, intersects the parabola \[{{x}^{2}}=2y\] at the point P in the first quadrant. Let the tangent to the circle \[{{C}_{1}}\]at P touches other two circles \[{{C}_{2}}\] and \[{{C}_{3}},\] at \[{{R}_{2}}\]and \[{{R}_{3}},\] respectively. Suppose \[{{C}_{2}}\], and \[{{C}_{3}},\]have equal radii \[2\sqrt{3}\]and centers \[{{Q}_{2}}\] and \[{{Q}_{3}}\], respectively. If \[{{Q}_{2}}\] and \[{{Q}_{3}},\] lie on the y-axis, then

A)  \[{{R}_{2}}{{R}_{3}}=4\sqrt{6}\]

B)  Area of the triangle \[O{{R}_{2}}{{R}_{3}}\] is \[6\sqrt{2}\]

C)  Area of the triangle \[P{{Q}_{2}}{{Q}_{3}}\] is \[6\sqrt{2}\]

D)  All of these

E)  None of these