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I have to describe geometrically sets of the form:

$\{z\in\mathbb{C} : Az \bar z +Bz + \bar B\bar z + C=0\}$,

$A,C\in\mathbb{R}, B\in\mathbb{C}$ .

Check also for $A=0$.

(I have a feeling that it is supposed to describe circles in the complex plane, but I don't have the knowledge to check it nor prove it)

Question posed by the professor for home exercise, on hyperbolic geometry on the complex upper half plane model course.

Nikolaos Kokkalis
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    Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or closed. To prevent that, please [edit] the question. This will help you recognize and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Jan 04 '22 at 00:38
  • What have you tried and where did you get stuck? If $A=0$ the LHS should remind you of a basic property, otherwise you can assume WLOG that $A=1$ and try to group the terms in $z, \bar z$ into a more familiar expression. – dxiv Jan 04 '22 at 00:43
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    I am extremely surprised that a student studying hyperbolic geometry in $\mathbb C$ of all things, would not even think to try the most basic technique of substituting $z = x + yi$ and proceed from there. Were I the professor, I would suspect that such a student lacks the prerequisite knowledge to take such a course. – heropup Jan 04 '22 at 01:26
  • I do actually lack the prerequisite knowledge – Nikolaos Kokkalis Jan 04 '22 at 01:28
  • @NikolaosKokkalis Alt. hints: if $A=0$ then $Bz+ \overline{,Bz,}+C = 0$, otherwise if $A=1$ then $z \bar z+ B z + \bar B \bar z \color{red}{+B \bar B - B\bar B}+C=0$. – dxiv Jan 04 '22 at 01:32

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