A circle C belongs to the family of circles passing through the intersection of the circle x^2 + y^2 - 4 = 0 and the line x + y - 2 = 0 . If a diameter of the circle C is a chord of the circle x^2 + y^2 + 8x + 8y + 4 = 0 , then the sum of the squares of the radii of all such possible circles C is

Step-by-step solution

The equation of the family of circles passing through the intersection of the given circle and line is: x^2 + y^2 - 4 + (x + y - 2) = 0 x^2 + y^2 + x + y - (2 + 4) = 0 The center of this circle C is (- 2 , - 2 ) . The square of its radius is R^2 = (- 2 )^2 + (- 2 )^2 - (-(2 + 4)) = ^2 2 + 2 + 4 . The given second circle is S₂ x^2 + y^2 + 8x + 8y + 4 = 0 . The common chord of circle C and circle S₂ is given by C - S₂ = 0 : ( - 8)x + ( - 8)y - 2 - 8 = 0 Since a diameter of circle C is a chord of S₂ , the common chord