Define coaxial circles and deduce their equation in simplest form.

A system of circles is said to be co-axial if every pair of the system has the same radical axis.
Circles passing through two fixed points form a coaxial system. The equation $S+\lambda P=0,S_1+\lambda S_2=0$ represent a family of coaxial circles.
Equation of coaxial circles in simplest form:
Let the common radical axis be chosen along y-axis and the line of centres which will be perpendicular to radical axis be along x-axis.
Hence the equation of any circle will be
$x^2+y^2+2gx+c=0.....\left(1\right)$
as y-co-ordinate of centre is zero.
Let any other circle of the system be
$x^2+y^2+2g_{1}x+2f_{1}y+c_1=0....\left(2\right)$
The radical axis of $\left(1\right)$ and $\left(2\right)$ is
$2x(g-g_1)-2f_{1}y+(c-c_1)=0....\left(3\right)$
But we are given that radical axis is y-axis
i.e. $x=0....\left(4\right)$.
Comparing $\left(3\right)$ and $\left(4\right)$, we get
$f_1=0$ and $c-c_1=0\therefore c_1=c$.
Hence any other circle of the system will have its equation of the form
$x^2+y^2+2g_{1}x+c=0$.
Thus the system of circles
$x^2+y^2+2g_{r}x+c=0$.
where $c$ is constant and $g_r$ a parameter represents a family of coaxial circles whose common radical axis is y-axis, i.e. $x=0$.