The equation ${\left|\mathrm{z}-1\right|}^2+{\left|\mathrm{z}+1\right|}^2=2$ represents 

$$\begin{gathered} {\left|\mathrm{z}-1\right|}^2+{\left|\mathrm{z}+1\right|}^2=2 \\ \mathrm{Let} \boxed{\mathrm{z}=\mathrm{x}+\mathrm{iy}}\Rightarrow {\left|\left(\mathrm{x}-1\right)+\mathrm{iy}\right|}^2+{\left|\left(\mathrm{x}+1\right)+\mathrm{iy}\right|}^2=2 \\ \Rightarrow {\left(\mathrm{x}-1\right)}^2+{\mathrm{y}}^2+{\left(\mathrm{x}+1\right)}^2+{\mathrm{y}}^2=2 \\ \Rightarrow 2{\mathrm{x}}^2+2{\mathrm{y}}^2+2=2 \\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2=0 \\ \therefore \mathrm{The} \mathrm{locus} \mathrm{represents} \left(\mathrm{x},\mathrm{y}\right)=\left(0,0\right) \end{gathered}$$