If and are complex numbers such that and

MATHEMATICS · Single correct · CBSE · Mathematics · Complex Numbers and Quadratic Equations · Complex Numbers and Quadratic Equations

Problem

z_1 = a + ib

and

z_2 = c + id

are complex numbers such that

|z_1| = |z_2| = 1

and

\operatorname{Re}(z_1 \bar{z}_2) = 0

, where

i = \sqrt{-1}

, then the complex numbers

\omega_1 = a + ic

and

\omega_2 = b + id

satisfy (1)

|\omega_1| = 1

(2)

|\omega_2| = 1

(3)

\operatorname{Re}(\omega_1 \bar{\omega}_2) = 0

(4) None of these