Verified Solution • 2025–26 Session moderate

QJEEM19Ma5b1d: JEE Main 2019 Mathematics - Class 11 JEEM19Ma5b1d Solution: If z and w are two complex numbers such that

If z and w are two complex numbers such that |zw| = 1 and arg(z) – arg(w) = π2{\pi \over 2} , then :

Multiple Choice Options

  • A zw‾=1−i2z\overline w = {{1 - i} \over {\sqrt 2 }}
  • B z‾w=i\overline z w = i
  • C zw‾=−1+i2z\overline w = {{ - 1 + i} \over {\sqrt 2 }}
  • D z‾w=−i\overline z w = -i

Step-by-Step Verified Solution

∣zw∣=1\left| {zw} \right| = 1

⇒\Rightarrow ∣z∣∣w∣=1\left| z \right|\left| w \right| = 1

Let w=1reiθw = {1 \over r}{e^{i\theta }}

then z = rei(θ+π2)r{e^{i\left( {\theta + {\pi \over 2}} \right)}}

z‾w=e−i(θ+π2).eiθ=e−i(π/2)=−i\overline z w = {e^{ - i\left( {\theta + {\pi \over 2}} \right)}}.{e^{i\theta }} = {e^{ - i(\pi /2)}} = - i

zw‾=ei(θ+π2).e−iθ=eiπ/2=iz\overline w = {e^{i\left( {\theta + {\pi \over 2}} \right)}}.{e^{ - i\theta }} = {e^{i\pi /2}} = i

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