If the equation a|z|^2+ z+ z+d=0 represents a circle, where a,d are real constants, then w…

Question

If the equation $a|z|^2+\alpha z+\bar\alpha\bar z+d=0$ represents a circle, where $a,d$ are real constants, then which of the following conditions is correct?

Accepted Answer

Writing $z=x+iy$, $\alpha=p+iq$, the equation reduces to $x^2+y^2+\dfrac{2p}ax+\dfrac{2q}ay+\dfrac da=0$, a genuine circle (radius$^2>0$) requires $a
e0$ and $\dfrac{p^2+q^2}{a^2}-\dfrac da>0$, i.e. $|\alpha|^2-ad>0$ with $a\in\mathbb R-\{0\}$.

Solutions · 1