Question

Let $\overline{z}$, $\overline{w}$ be complex numbers such that $z+w$ purely imaginary and $z-w$ is purely real

• A. $z=w$
• B. $z=-w$
• C. $z=\overline{w}$
• D. $z=-\overline{w}$

Answer 1

The correct option is D $z=-\overline{w}$

Let the complex number be $z=a+ib$

$z+w$ is purely imagnary so $Rew=-a$

$z-w$ is purely real So $Imw=ib$

$w=-a+ib-(a-ib)=-\overline{z}$