The circle C1 :` 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 C1 at P touches other two circles C2 and C3 at R2 and R3, respectively. Suppose C2 and C3 have equal radii `2sqrt(3)` and centers Q2 and Q3, respectively.If `Q_2 and Q_3` lies on the y-axis, then

Options

A. 3
B. 6
C. 9
D. 12

Answer

D

Explanation

On solving `x^(2)+y^(2)=3 and x^(2)=2y`, we find that the coordinates of point P are `(sqrt(2), 1)`. The equation of the tangent to the circle at `P(sqrt(2), 1)` is `sqrt(2)x+y=3`. Let the coordinates of the centre of the circles `C_(2) and C_(3)` be (0, k). The tangent at P i.e. `sqrt(2)x+y=3` also touches circle `C_(2) and C_(3)`.