Question

Let $z$ and $w$ be non-zero complex numbers such that $zw=\left|z^2\right|$ and $\left|z-\stackrel{-}{z}\right|+\left|w+\stackrel{-}{w}\right|=4.$ If $w$ varies, then the perimeter of the locus of $z$ is

• A. $8\sqrt{2}$ units
• B. $4\sqrt{2}$ units
• C. $8$ units
• D. $4$ units

Answer 1

Answer: (A)

Given, $zw={\left|z\right|}^2\Rightarrow zw=z\stackrel{-}{z}$
$\Rightarrow w=\stackrel{-}{z}\left\{z\ne 0\right\}$
Now, $\left|z-\stackrel{-}{z}\right|+\left|w+\stackrel{-}{w}\right|=4$
$\Rightarrow \left|z-\stackrel{-}{z}\right|+\left|z+\stackrel{-}{z}\right|=4$
Let, $z=x+iy$ , then we get,
$\left|x\right|+\left|y\right|=2$
which represents a square of side length equal to $2\sqrt{2}$

$\Rightarrow$ The perimeter of the locus is $8\sqrt{2}$ units