Let two intersecting circles be represented by equations x^2 + y^2 + Ax + By + C =0 and x^2 + y^2 + ax + by + c = 0.

For any number k that is not equal to -1, show that the equation of the circle through the intersection points of the two circles is x^2 + y^2 + Ax + By + C + k(x^2 + y^2 + ax + by + c) = 0.

Conversely, also show that every such circle may be represented by such an equation with a suitable k.