The circle `C_(1) : x^(2)+y^(2)=3`, with cenre 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 `2sqrt(3)` and centres `Q_(2)` and `Q_(3)` respectively. If `Q_(2)` and `Q_(3)` lie on the y-axis, then `Q_(2)Q_(3)=`

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)`.