Question

If $z$ is a non-zero complex number, then the area of the quadrilateral formed by the points $z,\overline{z},-z$ and $-\overline{z}$ is

• A. $|z{|}^2$
• B. $4|z{|}^2$
• C. $|z^2+{\overline{z}}^2|$
• D. $|z^2-{\overline{z}}^2|$

Answer 1

The correct option is D $|z^2-{\overline{z}}^2|$

Let $z=x+iy$
$\Rightarrow z^2=x^2-y^2+2ixy$
$\Rightarrow {\overline{z}}^2=x^2-y^2-2ixy$


Area $=\left(2x\right)\left(2y\right)=4xy=|z^2-{\overline{z}}^2|$