Question

If $\left|z-5i\right|=\left|z+5i\right|$, then find the locus of z.

Answer 1

$$\begin{gathered} \left|z-5i\right|=\left|z+5i\right| \\ \Rightarrow {\left|z-5i\right|}^2={\left|z+5i\right|}^2 \\ \Rightarrow \left(z-5i\right)\left(\overline{z-5i}\right)=\left(z+5i\right)\left(\overline{z+5i}\right)\left[\because z\overline{z}={\left|z\right|}^2\right] \\ \Rightarrow \left(z-5i\right)\left(\overline{z}+5i\right)=\left(z+5i\right)\left(\overline{z}-5i\right) \\ \Rightarrow z\overline{z}+5zi-5\overline{z}i-25i^2=z\overline{z}-5zi+5\overline{z}i-25i^2 \\ \Rightarrow 5zi+5zi=5\overline{z}i+5\overline{z}i \\ \Rightarrow 10zi=10\overline{z}i \\ \Rightarrow z=\overline{z} \\ \Rightarrow z\mathrm{is}\mathrm{purely}\mathrm{real} \end{gathered}$$

Hence, the locus of z is real axis.