If z and ω are two complex numbers such that |z ω|=1 and arg (z)- arg (ω)=(3 π/2), then arg ((1-2 barz ω/1+3 z ω)) is: (Here arg (z) denotes the principal argument of complex number z )

Question

If z and ω are two complex numbers such that |z ω|=1 and arg (z)- arg (ω)=(3 π/2), then arg ((1-2 barz ω/1+3 z ω)) is: (Here arg (z) denotes the principal argument of complex number z ) (A) (π/4) (B) -(3 π/4) (C) -(π/4) (D) (3 π/4)

Tardigrade Admin · Accepted Answer

As |z ω|=1 ⇒| f | z |=r, then |ω|=(1/r) Let arg (z)=θ ∴ arg (ω)=(θ-(3 π/2)) So, z=r ei θ ⇒ barZ=r ei(-θ) ω=(1/r) ei(θ-(3 π/2)) Now, consider (1-2 barz ω/1+3 barz ω)=(1-2 ei(-(3 π/2))/1+3 e(-(3 π)2))=((1-2 i/1+3 i)) =((1-2 i)(1-3 i)/(1+3 i)(1-3 i))=-(1/2)(1+i) ∴ prin arg ((1-2 barz ω/1+3 barz ω)) = prin arg ((1-2 barz ω/1+3 barz ω)) =(-(1/2)(1+i)) =-(π-(π/4))=(-3 π/4)

Q. If $z$ and $ω$ are two complex numbers such that $∣ z ω ∣ = 1$ and $ar g (z) - ar g (ω) = \frac{3 π}{2}$ , then $ar g (\frac{1 - 2 z ˉ ω}{1 + 3 z ω})$ is :
(Here $ar g (z)$ denotes the principal argument of complex number $z$ )

$\frac{π}{4}$

$- \frac{3 π}{4}$

$- \frac{π}{4}$

$\frac{3 π}{4}$

Q. If z and ω are two complex numbers such that ∣zω∣=1 and arg(z)−arg(ω)=23π​, then arg(1+3zω1−2zˉω​) is : (Here arg(z) denotes the principal argument of complex number z )

4π​

−43π​

−4π​

43π​

Q. If $z$ and $ω$ are two complex numbers such that $∣ z ω ∣ = 1$ and $ar g (z) - ar g (ω) = \frac{3 π}{2}$ , then $ar g (\frac{1 - 2 z ˉ ω}{1 + 3 z ω})$ is :
(Here $ar g (z)$ denotes the principal argument of complex number $z$ )

$\frac{π}{4}$

$- \frac{3 π}{4}$

$- \frac{π}{4}$

$\frac{3 π}{4}$