Question

If $z+\overline{z}=0$, then

• A. Re (z) $=0$
• B. Im (z) $=0$
• C. Re(z) $=$ Im(z) $=0$
• D. Im(z) $=$ Re(z)

Answer 1

The correct option is A Re (z) $=0$

Let $z=r{e}^{i\theta }$
Then $z+\overline{z}=0$
$\Rightarrow r{e}^{i\theta }+r{e}^{-i\theta }=0$
$\Rightarrow r(e^{i\theta }+e^{-i\theta })=0$
Since $r\ne 0$, we have
$e^{i\theta }+e^{-i\theta }=0$
$\cos \theta +i\sin \theta +(\cos \theta -i\sin \theta )=0$
$\left(2\cos \theta \right)=0$
$\cos \theta =0$
Thus $Re\left(z\right)=0$
Hence, $z$ is purely imaginary.