For two non-zero complex number $z_1$ 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. $\operatorname{Im}\left(z_1\right)0$ C. $\operatorname{Im}\left(z_1\right)0$ and $\operatorname{Im}\left(z_2\right)

Question

For two non-zero complex number $z_1$ 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: [JEE Main 2023, 29 Jan (Shift 1)]

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For two non-zero complex number \(z_1\) and \(z_2\), if \(\operatorname{Re}\left(z_1 z_2\right)\) \(=0\) and \(\operator…

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