Coaxial system of circles

The equation x² + y²+2gx+c = 0, where g is a variable and c is a constant is the simplest equation of a coaxial system of circles. The common radial adxis of this system of circles is y-axis.

If the equation of one of the circles and the radical axis are given:
Circle x² + y²+2gx+2fy+c = 0
Radical axis P = lx+my+n = 0

Then S+ λP = 0 (λ is an arbitrary constant) represents the coaxial system of circles.

If the equations of two of the circles are given

Circle 1 (termed as S1) x² + y²+2g1x+2f1y+c = 0

Circle 2 (termed as S2) x² + y²+2g2x+2f2y+c = 0

Then S1+λS2 = ) (λ ≠-1) represents the coaxial system.