If $\mathrm{z}=\mathrm{x}+\mathrm{iy}$ is a complex numbers z for which ${\left|\mathrm{z}+\mathrm{i}/2\right|}^2={\left|\mathrm{z}-\mathrm{i}/2\right|}^2$ then the locus of z is

$$\begin{gathered} {\left|\mathrm{z}+\frac{\mathrm{i}}{2}\right|}^2={\left|\mathrm{z}-\frac{\mathrm{i}}{2}\right|}^2 \\ \Rightarrow {\left|\mathrm{x}+\mathrm{iy}+\frac{\mathrm{i}}{2}\right|}^2={\left|\mathrm{x}+\mathrm{iy}-\frac{\mathrm{i}}{2}\right|}^2 \\ \Rightarrow {\left|\mathrm{x}+\mathrm{i}(\mathrm{y}+\frac{1}{2})\right|}^2={\left|\mathrm{x}+\mathrm{i}(\mathrm{y}-\frac{1}{2})\right|}^2 \\ \Rightarrow {\mathrm{x}}^2+{(\mathrm{y}+\frac{1}{2})}^2={\mathrm{x}}^2+{(\mathrm{y}-\frac{1}{2})}^2 {\mathrm{y}}^2+\mathrm{y}+\frac{1}{4}={\mathrm{y}}^2-\mathrm{y}+\frac{1}{4} \\ \therefore 2\mathrm{y}=0 \Rightarrow \mathrm{y}=0 (i.e) \boxed{\mathrm{x}-axis.} \end{gathered}$$