I feel like this problem given to me is much easier than I'm making it out to be.
Basically I'm only asked to write the equation of a circle with a given centre $(a,b)$, then the projective equation of the circle with the same centre. How can I do this?
Ordinary (affine) equation: $(x-a)^2+(y-b)^2=r^2$.
Substitute formally $x/z$ and $y/z$ instead of $x$ and $y$ and you will get the projective equation:
$$(x/z-a)^2+(y/z-b)^2=r^2$$
or (after formally multiplying by $z^2$):
$$(x-az)^2+(y-bz)^2=r^2z^2$$