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Q.

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 $\bar{z} w =$

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a

$- i$

b

1

c

$- 1$

d

$i$

answer is D.

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Detailed Solution

$z = r_{1} e^{i θ_{1}} z_{2} = r_{2} e^{i θ_{2}} | z w | = 1 \Rightarrow r_{1} r_{2} = 1 \begin{array}{l} A r g z - A r g w = \frac{π}{2} \Rightarrow θ_{1} - θ_{2} = \frac{π}{2} \\ \bar{z} w = r_{1} e^{- i θ_{1}} \times r_{2} e^{i θ_{2}} = r_{1} r_{2} e^{- i (θ_{1} - θ_{2})} \\ = 1 \times e^{- i \frac{π}{2}} = - i \end{array}$

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