The equation of an arbitrary circle includes three independent constants just as you give- and that means you will need a third degree equation to get those arbitrary constant as "constants of integration".
Just differentiating , with respect to x, gives and that has already eliminated a. Differentiating again, and we have eliminated a. How can you eliminate b?
You can get the equation of a circle passing through the origin by putting x=0, y=0 into the general equation:
so that the equation of a circle passing through the origin is
which has two arbitrary constants.