For two non-zero complex numbers z_1 and z_2, if Re(z_1z_2)=0 and Re(z_1+z_2)=0, then whic…

Question

For two non-zero complex numbers $z_1$ and $z_2$, if $\operatorname{Re}(z_1z_2)=0$ and $\operatorname{Re}(z_1+z_2)=0$, then which of the following are possible? (A) $\operatorname{Im}(z_1)>0$ and $\operatorname{Im}(z_2)>0$ (B) $\operatorname{Im}(z_1)<0$ and $\operatorname{Im}(z_2)>0$ (C) $\operatorname{Im}(z_1)>0$ and $\operatorname{Im}(z_2)<0$ (D) $\operatorname{Im}(z_1)<0$ and $\operatorname{Im}(z_2)<0$

Accepted Answer

Let $z_1=x_1+iy_1,\ z_2=x_2+iy_2$. Then:
$$\operatorname{Re}(z_1z_2)=x_1x_2-y_1y_2=0, \qquad \operatorname{Re}(z_1+z_2)=x_1+x_2=0 \implies x_2=-x_1.$$

Substituting: $-x_1^2-y_1y_2=0 \implies y_1y_2=-x_1^2\le0$.

So $y_1,y_2$ must have opposite signs (assuming $x_1\neq0$). This matches options (B) and (C).

Solutions · 1