Consider a family of circles passing through two fixed points A 3 , 7 and B 6 , 5 . The chord in which the circle x 2 + y 2 - 4 x - 6 y - 3 = 0 cuts each member of family of circles passes through a fixed point a , b . Then the value of a + 3 b is

Step-by-step solution

Family of circles is x - 3 x - 6 + y - 7 y - 5 + λ 2 x + 3 y - 27 = 0 i.e., x 2 + y 2 + x 2 λ - 9 + y 3 λ - 12 + 53 - 27 λ = 0 Common chord of family of circles and the given circle is - 5 x - 6 y + 56 + λ 2 x + 3 y - 27 = 0 which represents family of line passing through point of intersection of the lines − 5 x − 6 y + 56 = 0 and 2 x + 3 y − 27 = 0 The point of intersection is 2 , 23 3 a = 2 b = 23 3 a + 3 b = 2 + 23 = 25