= Classification of a real complex-circle equation
{title2=$A|z|^2+\overline Bz+B\overline z+C=0$}
For real $A,C$ and complex $B$, if $A\ne0$, <completing the square> gives $|z+B/A|^2=(|B|^2-AC)/A^2$. A positive right side gives a <circle>; zero gives a single point; a negative value gives the <empty set>. If $A=0$ and $B\ne0$, the equation is the <straight line> $2\operatorname{Re}(\overline Bz)+C=0$. If $A=B=0$, it describes the whole <complex plane> when $C=0$, and the <empty set> otherwise. Thus unrestricted coefficients require both the single-point and whole-plane degeneracies.
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