For two non-zero complex numbers z 1 and z 2 , if Re ( z 1 z 2 ) = 0 and Re ( z 1 + z 2 ) = 0 , then which of the following are possible? [2023]

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Solution

Q.
For two non-zero complex numbers $z_{1}$ and $z_{2}$ , if $Re (z_{1} z_{2}) = 0$ and $Re (z_{1} + z_{2}) = 0$ , then which of the following are possible

(A) $Im (z_{1}) > 0$ and $Im (z_{2}) > 0$
(B) $Im (z_{1}) < 0$ and $Im (z_{2}) > 0$
(C) $Im (z_{1}) > 0$ and $Im (z_{2}) < 0$
(D) $Im (z_{1}) < 0$ and $Im (z_{2}) < 0$

Choose the correct answer from the options given below: [2023]

1 A and B

2 B and C

3 B and D

4 A and C

Ans.
(2)

Let $z_{1} = x_{1} + i y_{1}$ ; $z_{2} = x_{2} + i y_{2}$

$Re (z_{1} z_{2}) \Rightarrow x_{1} x_{2} - y_{1} y_{2} = 0 ...(i)$

$Re (z_{1} + z_{2}) \Rightarrow x_{1} + x_{2} = 0 ...(ii)$

$\Rightarrow x_{1}^{2} + y_{1} y_{2} = 0 \Rightarrow y_{1} y_{2} = - x_{1}^{2}$

So, $Im (z_{1})$ and $Im (z_{2})$ are opposite in sign

So, the correct statements are B and C.

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