Let us fix $a,b,c\in\Bbb C$ and $r>0$.
The set of complex numbers $$ A:=\left\{z\in\Bbb C\;:\;\left|\frac b{z-c}-a\right|=r\right\} $$ is circle if $|a|\neq r$, a line otherwise.
In fact $A=f(C)$ where $C$ is the circle $\{w\in\Bbb C\;:\;|w-a|=r\}$ and $f$ is the Möbius transformations $f(z)=\frac b{z-c}$, and it's well known that circles/lines are preserved under MT.
In the case of $A$ being a circle, does there exist general formulas to express its center and radius?
Note $\left|\frac b{z-c}-a\right|^2=r^2$, or
$$\left( \frac b{z-c}-a\right) \left( \frac{ \bar{b}}{\bar{z}-\bar{c}}-\bar{a}\right) =r^2$$ Expand to get
$$|z|^2 -\left(c-\frac{\bar{a}b}{|r|^2-|a|^2}\right)\bar{z} - \left(\bar{c}-\frac{a\bar{b}}{|r|^2-|a|^2}\right)z = \frac{|b|^2}{|r|^2-|a|^2} + \frac{a\bar{b}c + \bar{a}b\bar{c}}{|r|^2-|a|^2} - |c|^2$$
Then, express the equation in the standard form of a circle
$$ \bigg| z-\left(c-\frac{\bar{a}b}{|r|^2-|a|^2}\right)\bigg|^2= \frac{|b|^2r^2}{(|r|^2-|a|^2)^2} $$
Thus, the center and the radius of the circle are respectively $$c-\frac{\bar{a}b}{|r|^2-|a|^2},\>\>\>\>\>\>\>\frac{|b|r}{||r|^2-|a|^2|}$$