Find the equation of the circle which passes through the point ( 1 , 2 ) and the points of intersection of the circles x 2 + y 2 - 8 x - 6 y + 21 = 0 and x 2 + y 2 - 2 x - 15 = 0 .

Step-by-step solution

Let S 1 = x 2 + y 2 - 8 x - 6 y + 21 = 0   &   S 2 = x 2 + y 2 - 2 x - 15 = 0 Using family of circles S 1 + λ S 2 = 0 x 2 + y 2 - 8 x - 6 y + 21 + λ x 2 + y 2 - 2 x - 15 = 0 Since it passes through 1 , 2 1 + 4 - 8 - 6 2 + 21 + λ 1 + 4 - 2 1 - 15 = 0 26 - 20 + λ 5 - 17 = 0 6 + λ - 12 = 0 λ = 1 2 So, the equation is x 2 + y 2 - 8 x - 6 y + 21 + 1 2 x 2 + y 2 - 2 x - 15 = 0 ⇒ 3 x 2 + 3 y 2 - 18 x - 12 y + 27 = 0