In the xy-plane, the graph of x² + y² − 12x − 4y + 15 = 0 is a circle. The line 3x − 4y = 25 intersects this circle at two points, A and B. What is the length of segment AB?

Question

Accepted Answer

B. 8. Complete the square: x² − 12x + y² − 4y + 15 = 0 → (x − 6)² − 36 + (y − 2)² − 4 + 15 = 0 → (x − 6)² + (y − 2)² = 25, so the center is (6, 2) and r = 5. The distance from (6, 2) to 3x − 4y − 25 = 0 is |3(6) − 4(2) − 25|/√(3² + (−4)²) = |−15|/5 = 3. The perpendicular from the center to a chord bisects it, so half the chord is √(5² − 3²) = 4, and AB = 2(4) = 8. (A) is the half-chord 4 — the student found the leg of the right triangle but forgot to double it. (C) = 2√(5² + 3²) = 2√34 — the student added the distance squared instead of subtracting it, misplacing the hypotenuse of the radius–distance–half-chord right triangle. (D) is the diameter 2r = 10, obtained by assuming the line passes through the center (it does not, since the distance is 3, not 0).

Answer

A. 4 — You correctly found the half-chord, √(5² − 3²) = 4, but AB is the full chord, so double it to get 8.

Answer

C. 2√34 — You added the distance squared instead of subtracting it; the radius 5 is the hypotenuse, so half the chord is √(5² − 3²) = 4, not √(5² + 3²).

Answer

D. 10 — That's the diameter, which would be the chord length only if the line passed through the center — but the center (6, 2) is 3 units from the line, not 0.

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Answer: B. 8

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