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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 $\operatorname{Re}\left(z_1+z_2\right)=0$, 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 :
$B$ and $D$
$B$ and $C$
$A$ and $B$
$A$ and $C$
Solution
$z _1= x _1+ i y _1$
$z _2= x _2+ iy _2$
$\operatorname{Re}\left(z_1 z_2\right)=x_1 x_2-y_1 y_2=0$
$\operatorname{Re}\left(z_1+z_2\right)=x_1+x_2=0$
$x_1$ and $x_2$ are of opposite sign
$y_1$ and $y_2$ are of opposite sign
