Let a circle of radius 4 pass through the origin O, the points A (- 3 a, 0) and B (0,- 2 b) , where a and b are real parameters and a b 0 . Then the locus of the centroid of O A B is a circle of radius

Step-by-step solution

Circle of radius 4 passes through O and points A( - 3a , 0), B(0, - 2b ). Let center be C(h, k) on circle: h^2 + k^2 = 16 . Since A, B lie on circle: 3a^2 + 2 3a h + h^2 + k^2 = 16 gives h = - 3a 2 And h^2 + 2b + 2 2b k + k^2 = 16 gives k = - 2b 2 Centroid G = ( - 3a 3 , - 2b 3 ) . Setting x = - 3a 3 and y = - 2b 3 : a = 3x^2 and b = 9y^2 2 From h^2 + k^2 = 16 : 3a 4 + 2b 4 = 16 → 3a + 2b = 64 Substituting: 9x^2 + 9y^2 = 64 → x^2 + y^2 = 64 9 Radius = 8 3