Question

Previous year IIT jee Question
Chapter :- complex number and quadratic equation.​

Answer 1

EXPLANATION.

If z and ω are two non-zero complex number.

⇒ |zω| = 1.

⇒ arg(z) - arg(ω) = π/2.

As we know that,

⇒ |z₁ z₂| = |z₁| |z₂|.

⇒ |z| |ω| = 1.

⇒ |ω| = 1/|z|

Let we assume that,

⇒ |z| = r.

⇒ |ω| = 1/r.

Let we assume that,

⇒ arg(ω) = θ.

⇒ arg(z) - c = π/2.

⇒ arg(z) = π/2 + θ.

As we know that,

If z = x + iy is a complex number,

Then it is written as,

$\implies z = re^{(\iota \theta)}$

$\implies z = re^{\iota (\pi/2 + \theta)}$

$\implies \omega = \dfrac{1}{r} e^{\iota \theta}$

$\implies \bar{z} . \omega = re^{- \iota (\pi/2 + \theta)} . \dfrac{1}{r} e^{\iota \theta}$

$\implies e^{- \iota \theta } = cos \theta - \iota sin \theta$

$\implies e^{\iota(- \pi/2 - \theta + \theta)} = e^{- \iota (\pi/2)} = - \iota$

$\implies \bar{z}. \omega = - \iota$