Question

If $z$ is a complex number such that $\mathrm{Re}\left(z\right)=\mathrm{Im}\left(z\right)$, then

• A. $\mathrm{Re}\left(z^2\right)=0$
• B. $\mathrm{Im}\left(z^2\right)=0$
• C. $\mathrm{Re}\left(z^2\right)=\mathrm{Im}\left(z^2\right)$
• D. $\mathrm{Re}\left(z^2\right)=-\mathrm{Im}\left(z^2\right)$
• E. $z^2=0$

Answer 1

The correct option is A

$\mathrm{Re}\left(z^2\right)=0$

Explanation for the correct option.

Find the real and imaginary part of $z^2$.

Let $z=x+iy$ be a complex number. Its real part is $x$ and imaginary part is $y$.

It is given that $\mathrm{Re}\left(z\right)=\mathrm{Im}\left(z\right)$, so $x=y$.

Now find $z^2$.

$\begin{array}{rcl}{z}^2& =& {\left(x+iy\right)}^2\\ & =& x^2+2x\times \left(iy\right)+i^2{y}^2\\ & =& x^2+2xyi+(-1)x^2\left[i^2=-1,x=y\right]\\ & =& x^2+2xyi-x^2\\ & =& 2xyi\end{array}$

So the complex number $z^2$ is $2xyi$. So its real part is $0$, $\mathrm{Re}\left(z^2\right)=0$ and its imaginary part is $2xy$, $\mathrm{Im}\left(z^2\right)=2xy$.

Hence, the correct option is A.