(D) Let $z = r_1 e^{i\theta_1}$ and $w = r_2 e^{i\theta_2}$.Given $|zw| = |z||w| = r_1 r_2 = 1$,so $r_2 = \frac{1}{r_1}$.Given $\arg(z) - \arg(w) = \t

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(D) Let $z = r_1 e^{i\theta_1}$ and $w = r_2 e^{i\theta_2}$.Given $|zw| = |z||w| = r_1 r_2 = 1$,so $r_2 = \frac{1}{r_1}$.Given $\arg(z) - \arg(w) = \theta_1 - \theta_2 = \frac{\pi}{2}$,so $\theta_1 = \theta_2 + \frac{\pi}{2}$.Now,consider $\bar{z}w = (r_1 e^{-i\theta_1})(r_2 e^{i\theta_2}) = (r_1 r_2) e^{i(\theta_2 - \theta_1)}$.Substituting the values,$\bar{z}w = (1) e^{i(-\pi/2)} = \cos(-\pi/2) + i\sin(-\pi/2) = -i$.Thus,$\bar{z}w = -i$.

Class 11 Mathematics 4-1.Complex numbers Mix Examples-Complex numbers

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If $z$ and $w$ are two complex numbers such that $|zw| = 1$ and $\arg(z) - \arg(w) = \frac{\pi}{2}$,then

JEE MAIN 2019 All JEE MAIN

A
$\bar{z}w = i$
B
$z\bar{w} = \frac{-1 + i}{\sqrt{2}}$
C
$z\bar{w} = \frac{1 - i}{\sqrt{2}}$
D
$\bar{z}w = -i$

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