Define coaxial circles and deduce their equation in the simplest form.

Step I: Define coaxial circles.

A family of circles, out of which every pair has the same radical axis, are called co-axial circles.

Step II: Deduce the equation of coaxial circles in its simplest form:

Let the radical axis that is common be along the axis of $y$ and the perpendicular lines called the line of centres to the radical axis be along the axis of $x$.

The equation of a circle will be $x^2+y^2+2gx+c=0$ — $\left(1\right)$

Since the$y-$coordinate of the 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

$2x(g–g_1)–2f_{1}y+(c–c_1)=0$ — $\left(3\right)$

The radical axis is the $y-$axis that is $x=0$…. $\left(4\right)$.

On comparing $\left(3\right)$ and $\left(4\right)$, $f_1=0$ and $(c–c_1)=0$.

Therefore, $c=c_1$

Thus, the system of circles $x^2+y^2+2g_{r}x+c=0$ where $c$ is constant and $g$ being a parameter represents a family of coaxial circles whose common radical axis is the $y-axis$, i.e. $x=0$.

Thus, the simplest form of the coaxial circles is $x^2+y^2+2g_{r}x+c=0$.