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  4. If z and w are non-zero complex numbers such that z w=|z|2, then |z- barz|+|w+ barw|=4 represents a

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Q. If $z$ and $w$ are non-zero complex numbers such that $z w=|z|^2$, then $|z-\bar{z}|+|w+\bar{w}|=4$ represents a

Complex Numbers and Quadratic Equations

Solution:

$z w=|z|^2 \Rightarrow z w=z \bar{z} \Rightarrow w=\bar{z} $
$\text { Thus, }|z-\bar{z}|+|w+\bar{w}|=4 $
$ \Rightarrow|z-\bar{z}|+|\bar{z}+z|=4 $
$ \Rightarrow|2 i y|+|2 x|=4 $
$ \Rightarrow|x|+|y|=2$
image
This represents a square. See Fig. 2.60.