If z and omega are two non-zero complex numbers such that |zomega |, = 1 and arg(z)

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4-1.Complex numbers

medium

If $z$ and $\omega $ are two non-zero complex numbers such that $|z\omega |\, = 1$ and $arg(z) - arg(\omega ) = \frac{\pi }{2},$ then $\bar z\omega $ is equal to

$1$

$-1$

$i$

$-i$

(AIEEE-2003)

Solution

(d) $|z|\,|\omega |\, = 1$ ……$(i)$
and $arg\,\left( {\frac{z}{\omega }} \right) = \frac{\pi }{2}\,\,\, \Rightarrow \,\,\frac{z}{\omega } = i$ $⇒$ $\left| {\frac{z}{\omega }} \right| = 1$ …..$(ii)$
From equation $(i)$ and $(ii)$
$|z|\, = \,|\omega |\, = 1$ and $\frac{z}{\omega } + \frac{{\bar z}}{{\bar \omega }} = 0;\,\,\,z\bar \omega + \bar z\omega = 0$
$\bar z\omega = – z\bar \omega = \frac{{ – z}}{\omega }\bar \omega \,\omega $; $\bar z\omega = – \,i\,|\omega {|^2} = – i.$.

Standard 11

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Solution

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