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Q.
If $|z+\bar{z}|=|z-\bar{z}|$, then the locus of $z$ is
Complex Numbers and Quadratic Equations
Solution:
Let $z=x+i y$. Then,
$|z+\bar{z}|=|z-\bar{z}|$
$\Leftrightarrow|2 x|=|2 i y| $
$\Rightarrow |x|=|y| \Leftrightarrow x=\pm y$
Hence, the locus of $z$ is a pair of straight lines.