Question

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

• A. rectangle
• B. square
• C. rhombus
• D. trapezium

Answer 1

Answer: (B)

$zw=|z{|}^2\Rightarrow zw=z\bar{z}\Rightarrow w=\bar{z}$
$\text{ Thus, }|z-\bar{z}|+|w+\bar{w}|=4$
$\Rightarrow |z-\bar{z}|+|\bar{z}+z|=4$
$\Rightarrow |2iy|+|2x|=4$
$\Rightarrow |x|+|y|=2$

This represents a square. See Fig. 2.60.