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Let A(5, -1), B(-2, 3), and C(-4, -7) be the vertices of triangle ABC. Let O be the origin. Six circles S_A, S_B, S_C, S_AB, S_AC, S_BC have diameters OA, OB, OC, AB, AC, and BC respectively. Identify the orthogonal pairs of circles from the options.

1. (S_A, S_BC)
2. (S_A, S_AC)
3. (S_B, S_AC)
4. No such orthogonal pair exists

Correct answer: (S_A, S_BC)

Solution

For S_A (centre A/2 = (5/2, -1/2), radius sqrt(26)/2) and S_BC (centre midpoint BC = (-3, -2), radius sqrt(26)): r1² + r2² = 26/4 + 26 = 130/4, and d² = (5/2+3)² + (-1/2+2)² = (11/2)² + (3/2)² = 130/4. They are orthogonal.

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