A line segment AB is bisected at C and semi-circles S1, S2 and S3 are drawn respectively on AB, AC and CB as diameters such that they all lie on same side of AB. A circle S is drawn touching internally S1 and externally S2 and S3. If r is the radius of S and R is the radius of S2, then which one of the following is correct?
- AR = 3r
- BR = 2r
- C3R = 4r
- D2R = 3r
Explanation
Place C at the origin with AB on the x-axis; let AC = CB = 2k so that S₁ has radius 2k (centre C) and S₂, S₃ have radius k. By symmetry the inscribed circle S has centre (0, y) with y = 2k − r (internal tangency to S₁) and distance √(k² + y²) = k + r (external tangency to S₂). Substituting: (2k − r)² + k² = (k + r)², which expands to 4k² − 4kr + r² + k² = k² + 2kr + r², i.e. 4k² = 6kr, so r = 2k/3. Since R = k (radius of S₂), R = 3r/2, i.e. 2R = 3r. The answer is (D).