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If $z$ and $w$ are two non-zero complex numbers such that $| z w | = 1$ , and $A r g (z) - A r g (w) = \frac{π}{2},$ then $\bar{z} ω$ is equal to

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-i

answer is D.

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Detailed Solution

$z = r_{1} {e^{i α}}^{} a n d w = r_{2} {e^{{i β}^{}}}^{}$ and $\bar{z} = r_{1} e^{- i α}$

$|z w| = 1$ $\Rightarrow r_{1} r_{2} = 1$

$a r g z - a r g w = \frac{π}{2}$ $\Rightarrow α - β = \frac{π}{2}$

$\bar{z} w = r_{1} r_{2} e^{- i (α - β)}$ $= - i$

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