If z and w are two complex numbers such that |zw| = 1 and… — JEE Main 2019 Mathematics PYQ

JEE Main 2019 · 10 April · Shift II

z

and

w

are two complex numbers such that

|zw| = 1

and

\arg(z) - \arg(w) = \dfrac{\pi}{2}

, then

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

Show answer and solution

Correct option: A

Solution

Let z = r·e^(iα) and w = s·e^(iβ). Then |zw| = rs = 1, and arg(z) - arg(w) = α - β = π/2.

z̄w = r·e^(-iα) · s·e^(iβ) = rs · e^(i(β-α)) = 1 · e^(-iπ/2) = cos(-π/2) + i·sin(-π/2) = -i.

[JEE Main 2019 · 10 April · Shift II]

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JEE Main 2019 Mathematics question on Complex Numbers and Quadratic Equations