Question Detail

Let the circle equation be x² + y² + Dx + Ey + F = 0. Substituting the three points:(1,5): 1 + 25 + D + 5E + F = 0 → D + 5E + F = -26(5,3): 25 + 9 + 5D + 3E + F = 0 → 5D + 3E + F = -34(3,-1): 9 + 1 + 3D - E + F = 0 → 3D - E + F = -10Subtract first from second: 4D - 2E = -8 → 2D - E = -4Subtract third from second: 2D + 4E = -24 → D + 2E = -12Solve: From 2D - E = -4 → E = 2D + 4. Substitute into D + 2(2D+4) = -12 → D + 4D + 8 = -12 → 5D = -20 → D = -4. Then E = 2(-4)+4 = -8+4 = -4. Then from D + 5E + F = -26: -4 + 5(-4) + F = -26 → -4 -20 + F = -26 → F = 2. So equation: x² + y² - 4x - 4y + 2 = 0. But that's not among options. Option D is x² + y² - 4x - 4y - 2 = 0, which would give F = -2. There might be a sign error.