Question

If $\left|z-\overline{z}\right|+\left|z+\overline{z}\right|=2$, then $z$ lies on

• A. a circle
• B. a square
• C. an ellipse
• D. a line

Answer 1

The correct option is B

a square

The explanation for the correct option:

Let us assume that $z=x+iy$, where $i=\sqrt{-1}$.

Thus, $\overline{z}=x-iy$.

It is given that, $\left|z-\overline{z}\right|+\left|z+\overline{z}\right|=2$

$$\begin{gathered} \Rightarrow \left|x+iy-x+iy\right|+\left|x+iy+x-iy\right|=2 \\ \Rightarrow \left|2iy\right|+\left|2x\right|=2 \\ \Rightarrow \sqrt{{\left(2y\right)}^2}+\sqrt{{\left(2x\right)}^2}=2 \\ \Rightarrow 2\left|x\right|+2\left|y\right|=2 \\ \Rightarrow \left|x\right|+\left|y\right|=1 \end{gathered}$$

As $\left|x\right|+\left|y\right|=1$ is an equation of a square, thus $z$ lies on a square.

Hence, (B) is the correct option.