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C1 and C2 are two concentric circles with radii r1 and r2 (r1 < r2). From any point P on C2, two tangents are drawn to C1, meeting C2 again at points A and B. If AB is always a diameter of C2, then which relation holds?

1. r2 = 2r1
2. r2 = √2 r1
3. r2² < 2r1²
4. C2 is director circle of C1

Correct answer: r2 = √2 r1

Solution

The tangent from P(r2,0) that meets C2 at A(0,r2) has equation y = -x + r2. Distance from center O to this line = r2/sqrt(2). For tangency to C1, this distance = r1, giving r1 = r2/sqrt(2), i.e., r2 = sqrt(2)*r1.

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