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Consider circles S1: x² + y² - 4x - 6y + 12 = 0 (center (2,3), radius 1) and S2: (x-5)² + (y-6)² = r² (center (5,6), radius r > 1). The distance between centers is 3*sqrt(2). Match each condition to the correct divisibility: (A) S1 and S2 touch internally: which of 3,4,5,6 divides (r-1)²? (B) S1 and S2 touch externally: which of 3,4,5,6 divides r² + 2r + 3? (C) S1 and S2 intersect orthogonally: which of 3,4,5,6 divides r² - 1? (D) Common chord of intersection is longest possible: which of 3,4,5,6 divides r² + 5?

1. 3
2. 4
3. 5
4. 6

Correct answer: 4

Solution

For case (C): orthogonal intersection gives d² = r1² + r² so r² = 17 and r² - 1 = 16, which is divisible by 4. This is the option most cleanly matched to a single answer from the given choices.

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