Question

If z and ω are two complex numbers such that |zω| = 1 and arg(z) – arg(ω) = `(3π)/2`, then `"arg"((1 - 2barzω)/(1 + 3barzω))` is ______. (Here arg(z) denotes the principal argument of complex number z)

Options

• `(3π)/4`

• `-π/4`

• `-(3π)/4`

• `π/4`

Answer 1

If z and ω are two complex numbers such that |zω| = 1 and arg(z) – arg(ω) = `(3π)/2`, then `"arg"((1 - 2bar"z"ω)/(1 + 3bar"z"ω))` is `underlinebb(π/4)`. (Here arg(z) denotes the principal argument of complex number z)

Explanation:

Given, |zω| = 1 and arg(z) – arg(ω) = `(3π)/2`

⇒ |z||ω| = 1 and arg(z) = arg(ω) + `(3π)/2`

Let ω = re^iθ ⇒ z = `1/"r""e"^("i"((3π)/2 + θ)`

⇒ `(1 - 2bar"z"ω)/(1 + 3bar"z"ω) = (1 - 2"r""e"^("i"θ) 1/"r""e"^("i"((3π)/2 - θ)))/(1 + 3"r""e"^("i"θ) 1/"r""e"^("i"((3π)/2 - θ)`

= `(1 - 2"e"^(-"i"((3π)/2)))/(1 + 3"e"^(-"i"((3π)/2)`

= `(1 - 2"i")/(1 + 3"i")`

So, `"arg"((1 - 2bar"z"ω)/(1 + 3bar"z"ω)) = "arg"((1 - 2"i")/(1 + 3"i"))`

= tan^–1(–2) – tan^–1(3)

= `π/4`