If z and ω are two non-zero complex numbers such that |zω| = 1 and Arg (z) - Arg (ω) = (π/2) , and barz ω is equal to ik, where k is smallest natural number then k is

Question

If z and ω are two non-zero complex numbers such that |zω| = 1 and Arg (z) - Arg (ω) = (π/2) , and barz ω is equal to ik, where k is smallest natural number then k is (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

| barz ω| = | barz||ω| = |z||ω| = |z ω |= 1 Arg ( barz ω) = arg ( barz) + arg(ω) = - arg (z) + arg ω = -(π/2) ∴ barz ω = -1

Q. If $z$ and $ω$ are two non-zero complex numbers such that $∣ z ω ∣ = 1$ and Arg $(z) -$ Arg $(ω) = \frac{π}{2}$ , and $\overset{z}{ˉ} ω$ is equal to $i^{k}$ , where $k$ is smallest natural number then $k$ is

$∣ \overset{z}{ˉ} ω ∣ = ∣ \overset{z}{ˉ} ∣∣ ω ∣ = ∣ z ∣∣ ω ∣ = ∣ z ω ∣ = 1$
Arg $(\overset{z}{ˉ} ω) =$ arg $(\overset{z}{ˉ}) +$ arg $(ω)$
$= -$ arg $(z) +$ arg $ω = - \frac{π}{2}$
$∴ \overset{z}{ˉ} ω = - 1$

Q. If z and ω are two non-zero complex numbers such that ∣zω∣=1 and Arg (z)− Arg (ω)=2π​, and zˉω is equal to ik, where k is smallest natural number then k is

∣zˉω∣=∣zˉ∣∣ω∣=∣z∣∣ω∣=∣zω∣=1 Arg (zˉω)= arg (zˉ)+ arg(ω) =− arg (z)+ arg ω=−2π​ ∴zˉω=−1

Q. If $z$ and $ω$ are two non-zero complex numbers such that $∣ z ω ∣ = 1$ and Arg $(z) -$ Arg $(ω) = \frac{π}{2}$ , and $\overset{z}{ˉ} ω$ is equal to $i^{k}$ , where $k$ is smallest natural number then $k$ is

$∣ \overset{z}{ˉ} ω ∣ = ∣ \overset{z}{ˉ} ∣∣ ω ∣ = ∣ z ∣∣ ω ∣ = ∣ z ω ∣ = 1$
Arg $(\overset{z}{ˉ} ω) =$ arg $(\overset{z}{ˉ}) +$ arg $(ω)$
$= -$ arg $(z) +$ arg $ω = - \frac{π}{2}$
$∴ \overset{z}{ˉ} ω = - 1$