<s>[INST] Collections of points of affixes $z$ Suppose a and b two different complex numbers and k a real positive number.
How can determinate  the collections of points z such as $ L_k=\{{z \in \mathbb{C}} / |\frac{z-a}{z-b}|=k \} $. 
 [/INST] If we rearrange the defining equation and square it, we get
$$\lvert z-a\rvert^2 = k^2\lvert z-b\rvert^2.$$
Expanding the squared moduli, $\lvert w\rvert^2 = w\overline{w}$, we obtain
$$\begin{gather}
(z-a)(\overline{z}-\overline{a}) = k^2 (z-b)(\overline{z}-\overline{b})\\
z\overline{z} - a\overline{z} - \overline{a}z + a\overline{a} = k^2(z\overline{z} - b\overline{z} - \overline{b}z + b\overline{b})\\
(1-k^2)z\overline{z} -(a-k^2b)\overline{z} -\overline{(a-k^2b)}z + \lvert a\rvert^2 - k^2\lvert b\rvert^2 = 0
\end{gather}$$
The last is an equation for a line if $k^2 = 1$ (since $k > 0$, that means $k = 1$), and an equation for a circle if $k^2 \neq 1$ and $\lvert a-k^2b\rvert^2 > (1-k^2)(\lvert a\rvert^2 - k^2\lvert b\rvert^2)$. The latter is always the case for $a \neq b$ and $k > 0$.
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