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 z ¯ ω 1 + 3 z ¯ ω is: (Here arg z denotes the principal argument of complex number z )

Step-by-step solution

As z ω = 1 If z = r , then ω = 1 r Let arg z = θ ∴ arg ω = θ - 3 π 2 So, z = r e i θ ⇒ z ¯ = r e i - θ ⇒ ω = 1 r e i θ - 3 π 2 Now, consider 1 - 2 z ¯ ω 1 + 3 z ¯ ω = 1 - 2 e i - 3 π 2 1 + 3 e i - 3 π 2 = 1 - 2 i 1 + 3 i = 1 - 2 i 1 - 3 i 1 + 3 i 1 - 3 i = 1 - 5 i + 6 i 2 10 = - 5 + 5 i 10 = - 1 2 1 + i Then, principal arg 1 - 2 z ¯ ω 1 + 3 z ¯ ω = principal arg - 1 2 1 + i = - π - π 4 = - 3 π 4