For two non-zero complex numbers z_1 and z_2, if Re (z_1 z_2)=0 and Re (z_1+z_2)=0, then which of the following are possible? A. Im (z_1)>0 and Im…

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