For two non-zero complex number z₁ and z₂, if Re (z₁z₂)=0 and Re (z₁+z₂)=0 then which of the following are possible?
A) Im(z₁)>0 and Im(z₂)>0
B) Im(z₁)<0 and Im(z₂)>0
C) Im(z₁)>0 and Im(z₂)<0
D) Im(z₁)<0 and Im(z₂)<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 A.

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Detailed Solution

Let $z_{1} = x_{1} + {iy}_{1}, z_{2} = n_{2} + {iy}_{2}$
$\begin{array}{l} x_{1} + x_{2} = 0 \Rightarrow x_{2} = - x_{1} \\ x_{1} x_{2} - y_{1} y_{2} = 0 \\ y_{1} y_{2} = - x_{1}^{2} < 0 \\ ∴ y_{1}, y_{2} have opp. signs \end{array}$

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