For two fixed complex numbers ${\mathrm{z}}_1,{\mathrm{z}}_2$ and a variable complex number z, the equation ${|\mathrm{z}-{\mathrm{z}}_1|}^2+{|\mathrm{z}-{\mathrm{z}}_2|}^2=2$ represents a circle if $|{\mathrm{z}}_1-{\mathrm{z}}_2|$ equals

$\begin{array}{l}{\left|z-z_1\right|}^2+{\left|z-z_2\right|}^2=k\\ \Rightarrow 2|z{|}^2+{\left|z_1\right|}^2+{\left|z_2\right|}^2-2\mathrm{Re}\left(z{\overline{z}}_2\right)-2\mathrm{Re}\left(z{\overline{z}}_1\right)=\mathrm{k}\\ \Rightarrow |\mathrm{z}{|}^2-\mathrm{Re}\left(\mathrm{z}\left({\overline{z}}_1+{\overline{z}}_2\right)\right)=\frac{1}{2}\left(\mathrm{k}-{\left|{\mathrm{z}}_1\right|}^2-{\left|{\mathrm{z}}_2\right|}^2\right)\\ \Rightarrow {\left|z-\frac{z_1+z_2}{2}\right|}^2-\frac{1}{4}{\left|z_1+z_2\right|}^2=\frac{1}{2}\left(k-{\left|z_1\right|}^2-{\left|z_2\right|}^2\right)\\ \Rightarrow {\left|z-\frac{z_1+z_2}{2}\right|}^2=\frac{1}{2}\mathrm{k}-\frac{1}{4}\left({\left|z_1\right|}^2+{\left|z_2\right|}^2-2\mathrm{Re}\left(z_1{\overline{z}}_2\right)\right)\end{array}$

$$\begin{gathered} \Rightarrow {\left|\mathrm{z}-\frac{{\mathrm{z}}_1+{\mathrm{z}}_2}{2}\right|}^2=\frac{1}{4}\left(2\mathrm{k}-{\left|{\mathrm{z}}_1-{\mathrm{z}}_2\right|}^2\right) \\ \text{which represents the real circle having } \end{gathered}$$

$\text{ centre at }\frac{z_1+z_2}{2}\text{ and radius is }\frac{1}{2}\sqrt{2\mathrm{k}-{\left|{\mathrm{z}}_1-{\mathrm{z}}_2\right|}^2}$

$\Rightarrow {\left|{\mathrm{z}}_1-{\mathrm{z}}_2\right|}^2\le 2\mathrm{k}$

If represents a circle if  $|{\mathrm{z}}_1-{\mathrm{z}}_2|\le 2$