For two non–zero complex number $z_{1}$ and $z_{2},$ if $Re (z_{1} z_{2}) = 0$ and $Im (z_{1} + z_{2}) = 0$ then which of the following are possible?
A) $Re (z_{1}) > 0$ and $Re (z_{2}) > 0$ B) $Re (z_{1}) < 0$ and $Re (z_{2}) > 0$
C) $Re (z_{1}) > 0$ and $Re (z_{2}) < 0$ D) $Re (z_{1}) < 0$ and $Re (z_{2}) < 0$
Choose the correct answer from the options given below

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B and D

A and B

A and C

B and C

answer is B.

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Detailed Solution

$Re (z_{1} z_{2}) = 0, Im (z_{1} + z_{2}) = 0$
$z_{1} = x_{1} + i y_{1}, z_{2} = x_{2} + i y_{2}$
$y_{1} + y_{2} = 0$ and $x_{1} x_{2} - y_{1} y_{2} = 0$
$y_{1} = - y_{2}$ and $x_{1} x_{2} = y_{1} y_{2}$
$x_{1} x_{2} < 0$
$\Rightarrow x_{1}, x_{2}$ are of opposite sign
$y_{1}, y_{2}$ are of opposite sign