Interpreting ${z\in \mathbb{C} : A\vert z \vert^2 - \bar{B}z + C = 0}$ geometrically?

Question

A: Given $A, C \in \mathbb{R}, A \neq 0, \vert B \vert^2 > A C$ geometrically characterize this set:

$$\{z\in \mathbb{C} : A\vert z \vert^2 - \bar{B}z + C = 0\}$$

I just can't grasp it no matter how many ways I try to rewrite it as.

Also, can I ask for help for these two other parts of the question? B. Calculate the equation of the circumference with $z_1, z_2, z_3 $ not lying on the same line.

C. Given $\vert a \vert \neq 0, 1$ Let's suppose 0, a, b not lying on the same line. Calculate the center and radius of the circumference that goes through the points: $a, b, 1/(\bar{a}$). Proove that $1/(\bar{b}$ is also on the circumference

Answer 1

Let $B=b_1+b_2i$, $b_1,b_2\in\Bbb R$, $z=x+i\,y$, $x,y\in\Bbb R$. Taking real and imaginary part in the equation defining the set, and using that $A,C$ are real, we get the following equations: $$ A(x^2+y^2)-(b_1x+b_2y)+C=0\tag{real part} $$

$$ b_1y-b_2x=0\tag{imaginary part} $$ The first one is a circle and the second one a straight line through the origin.