Circle C1: x² + y² = r² is fixed. Circle C2 is such that from every point on C2, two perpendicular tangents can be drawn to C1. A third circle C3 passes through the center of C1, intersects C2 at P and Q and intersects C1 at A and B such that line PQ is tangent to C1 at R. Find the locus o

Question

Circle C1: x² + y² = r² is fixed. Circle C2 is such that from every point on C2, two perpendicular tangents can be drawn to C1. A third circle C3 passes through the center of C1, intersects C2 at P and Q and intersects C1 at A and B such that line PQ is tangent to C1 at R. Find the locus of the center of variable circle C3.

Accepted Answer

C2. C2 is the director circle of C1 with equation x²+y² = 2r². PQ (chord of C2 intersection) is tangent to C1 at R, meaning the distance from origin O to PQ equals r. PQ is the radical axis of C3 and C2. The center of C3 lies on the radical axis of C1 and C3 (perpendicular bisector of AB). Through the geometric constraints that O lies on C3 and PQ is tangent to C1, the locus of the center of C3 is the director circle C2.

Solution

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