Equations \ of \ circles \ in \ \mathbb{C}
Suppose \ we \ have \ a \ point \ c\in\mathbb{C}
Let \ C \ be \ the \ circle \ with \ centre \ c and \ radius \ r.
Then \ the \ equation \ of \ C \ is
{(z-c)(\overline{z-c})=r^{2}}
{z \bar{z}-\bar{c} z-c \bar{z}+|c|^{2}-r^{2}=0}
Thus, \ we \ can \ say \ that \ the \ equation
{z \bar{z}-\bar{c} z-c \bar{z}+\lambda=0 \quad c\in\mathbb{C}, \lambda\in\mathbb{R}}
is \ the \ equation \ of \ a \ circle \ centered \ at \ c \ with \ radius \ \sqrt{|c|^{2}-\lambda}