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The circle C1: x² + y² = 3 (centre O) meets the parabola x² = 2y at a point P in the first quadrant. The tangent to C1 at P touches two other circles C2 and C3 (equal radii 2 sqrt(3), centres Q2 and Q3 on the y-axis) at R2 and R3. Which of the following are correct?

1. Q2Q3 = 12 and R2R3 = 4 sqrt(6)
2. Q2Q3 = 6 and R2R3 = 2 sqrt(6)
3. Q2Q3 = 12 and area of triangle OR2R3 = 6 sqrt(2)
4. Q2Q3 = 8 and R2R3 = 4 sqrt(6)

Correct answer: Q2Q3 = 12 and R2R3 = 4 sqrt(6)

Solution

Solving gives P = (sqrt 2, 1); the tangent line is sqrt2 x + y = 3. Setting distance from (0, c) to this line equal to 2 sqrt 3 gives c = 9 and c = -3, so Q2Q3 = 12 and R2R3 = 4 sqrt 6.

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