Complex Numbers

For two non-zero complex numbers \(z_{1}\) and \(z_{2}\), if \(\operatorname{Re}\left(z_{1} z_{2}\right)=0\) and \(\operatorname{Re}\left(z_{1}+z_{2}\right)=0\), then which of the following are possible?

A. \(\operatorname{Im}\left(z_{1}\right)>0\) and \(\operatorname{Im}\left(z_{2}\right) > 0\)

B. \(\operatorname{Im}\left(z_{1}\right) < 0\) and \(\operatorname{Im}\left(z_{2}\right) > 0\)

C. \(\operatorname{Im}\left(z_{1}\right) > 0\) and \(\operatorname{Im}\left(z_{2}\right) < 0\)

D. \(\operatorname{Im}\left(z_{1}\right) < 0\) and \(\operatorname{Im}\left(z_{2}\right) < 0\)

Choose the correct answer from the options given below :