Question

For two non-zero complex numbers $z_1$ and $z_2$, if $\text{Re}\left(z_1{z}_2\right)=0$ and $\text{Re}\left(z_1+z_2\right)=0$, then which of the following are possible?
A. $\text{Im}\left(z_1\right)>0$ and $\text{Im}\left(z_2\right)>0$
B. $\text{Im}\left(z_1\right)<0$ and $\text{Im}\left(z_2\right)>0$
C. $\text{Im}\left(z_1\right)>0$ and $\text{Im}\left(z_2\right)<0$
D. $\text{Im}\left(z_1\right)<0$ and $\text{Im}\left(z_2\right)<0$
Choose the correct answer from the options given below:

• A. A and B
• B. $B$ and $C$
• C. B and D
• D. A and C

Answer 1

Answer: (B)

$z_1=x_1+i{y}_1$
$z_2=x_2+i{y}_2$
$\mathrm{Re}\left(z_1{z}_2\right)=x_1{x}_2-y_1{y}_2=0$
$\mathrm{Re}\left(z_1+z_2\right)=x_1+x_2=0$
$x_1\&x_2$ are of opposite sign
$y_1\&y_2$ are of opposite sign