Question:
If $|z-5 i|=|z+5 i|$, then find the locus of $z$.
Solution:
$|z-5 i|=|z+5 i|$
$\Rightarrow|z-5 i|^{2}=|z+5 i|^{2}$
$\Rightarrow(z-5 i)(\overline{z-5 i})=(z+5 i)(\overline{z+5 i}) \quad\left[\because z \bar{z}=|z|^{2}\right]$
$\Rightarrow(z-5 i)(\bar{z}+5 i)=(z+5 i)(\bar{z}-5 i)$
$\Rightarrow z \bar{z}+5 z i-5 \bar{z} i-25 i^{2}=z \bar{z}-5 z i+5 \bar{z} i-25 i^{2}$
$\Rightarrow 5 z i+5 z i=5 \bar{z} i+5 \bar{z} i$
$\Rightarrow 10 z i=10 \bar{z} i$
$\Rightarrow z=\bar{z}$
$\Rightarrow z$ is purely real
Hence, the locus of $z$ is real axis.
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