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Complex Numbers

Let z1= a+i...

Question

Let $z_{1} = a + i b, z_{2} = p + i q$ be two unimodular complex numbers such that $I m (z_{1} z_{2}) = 1$ . If $ω_{1} = a + i p, ω_{2} = b + i q$ then

A

R e (ω_{1} ω_{2}) = 1

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B

I m (ω_{1} ω_{2}) = 1

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C

R m (ω_{1} ω_{2}) = 0

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D

I m (ω_{1}_{2}) = 0

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Solution

The correct option is A $I m (ω_{1} ω_{2}) = 1$
$z_{1} = a + i b$ and $z_{2} = p + i q$
Given that $I m (z_{1} z_{2}) = 1$ , so $a q + b p = 1$
Now, $ω_{1} = a + i p, ω_{2} = b + i q$
$\Rightarrow ω_{1} ω_{2} = a b - p q + i (a q + b p)$
$\Rightarrow I m (ω_{1} ω_{2}) = 1$

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