Question

If the four complex numbers $z$, $\overline{z},\overline{z}-2Re\left(\overline{z}\right)$ and $z-2Re\left(z\right)$ represent the vertices of a square of side $4$ units in the Argand plane, then $\left|z\right|$ is equal to:

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

Answer 1

The correct option is D $2\sqrt{2}$


Let vertices of given square are
$A\left(z\right),B\left(\overline{z}\right),C(\overline{z}-2Re(\overline{z}\left)\right),D(z-2Re(z\left)\right).$

Let $z=x+iy$
Now, $C{A}^2=A{B}^2+B{C}^2$
$\Rightarrow 32=2^2{y}^2+2^2{x}^2$
$\Rightarrow x^2+y^2=8$
$\therefore |z|=\sqrt{x^2+y^2}=2\sqrt{2}$