How do find out circles represent a family of coaxial circles from their equations? Coaxial Circles are circles whose centres are collinear and that share a common radical line. Given are two equations of circles $x^2+y^2+a^2+2gx=0$ $x^2+y^2-2fy-a^2=0$ they represent a family of coaxial Circles. How to know that?

All the circles in the same family pass through the same couple of points (either real or imaginary).

In your case it is easy to see that all circles represented by the first equation pass through $(0,\pm ia)$, while all circles represented by the second equation pass through $(\pm a,0)$. So, once the value of $a$ is fixed, they both represent families of coaxial circles parameterized by $g$ and $f$.

If common points are not so evident, one can find them by remembering that if an equation $F(x,y,k)=0$ is satisfied by the same point $(x,y)$ for every value of $k$, then that point is also a solution of $\partial f/\partial k=0$ for every value of $k$.