= Reflection symmetry of a complex circle equation
{title2=$z\mapsto (\beta/\bar\beta)\bar z$}
For real $\alpha\ne0,\gamma$ and complex $\beta\ne0$, completing the square in $\alpha|z|^2-\beta\bar z-\bar\beta z+\gamma=0$ gives center $\beta/\alpha$ and radius $\sqrt{|\beta|^2-\alpha\gamma}/|\alpha|$, when the radicand is nonnegative. The displayed map is reflection in the line through $0$ and $\beta$: it preserves the <complex modulus>, fixes the center, and squares to the identity. Hence it maps the <circle> onto itself, including a radius-zero degeneration.
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