Equation of a circle through the intersection of a circle and line

The equation of a circle passing through the intersection (points of intersection) of the circle S = x² + y²+2g1x+2f1y+c = 0 and the line L = lx+my+n = 0 is
x² + y²+2g1x+2f1y+c+ λ(lx+my+n) = 0 or
S+ λL = 0 where λ is a constant determined by an additional condition.