Circles

Any circle written as x^2 + y^2 + 2gx + 2fy + c = 0 has centre (-g, -f) and radius sqrt(g^2 + f^2 - c); identifying these coefficients is the first move in every circle problem of this type. Here 2g = -6 so g = -3, and 2f = 8 so f = 4, while c = -11. The centre is therefore (-g, -f) = (3, -4). The radius equals sqrt(g^2 + f^2 - c) = sqrt(9 + 16 + 11) = sqrt(36) = 6. Option Centre (-3, 4) flips the sign rule by forgetting the negation in (-g, -f). Option radius 36 mistakes the radicand for the radius itself. Option Centre (6, -8) reads the raw coefficients without halving. This is the standard JEE Advanced completing-the-square identification. Plausibility check: completing squares gives (x - 3)^2 + (y + 4)^2 = 36, confirming centre (3, -4) and radius 6, and the radicand 36 being a perfect square reassures the arithmetic.