If $z$ and $\omega$ are two complex numbers such that $|z \omega|=1 \operatorname{and} \arg (z)-\arg (w)=\frac{3 \pi}{2}$, then $\arg \left(\frac{1-2 \bar{z} \omega}{1+3 \bar{z} \omega}\right)$ is : (Here $\arg (\mathrm{z})$ denotes the principal argument of complex number z )

JEE Main · 2021 · 20 Jul 2021 · Shift 1 · Complex Numbers

z

and

ω

are two complex numbers such that

∣ z ω ∣ = 1 and ar g (z) - ar g (w) = \frac{3 π}{2}

, then

ar g (\frac{1 - 2 z ˉ ω}{1 + 3 z ˉ ω})

is :
(Here

ar g (z)

denotes the principal argument of complex number z )

Answer: (B)

The modulus of

\overset{z}{ˉ} ω

equals that of

z ω

, so

∣ \overset{z}{ˉ} ω ∣ = 1

, and

ar g (\overset{z}{ˉ} ω) = - ar g z + ar g ω = - \frac{3 π}{2} \equiv \frac{π}{2} (mod 2 π)

which makes

\overset{z}{ˉ} ω = i

. Substituting this value,

\frac{1 - 2 z ˉ ω}{1 + 3 z ˉ ω} = \frac{1 - 2 i}{1 + 3 i} = \frac{( 1 - 2 i ) ( 1 - 3 i )}{( 1 + 3 i ) ( 1 - 3 i )} = \frac{- 5 - 5 i}{10} = - \frac{1}{2} - \frac{i}{2}

Both the real and the imaginary part are negative, so the point lies in the third quadrant with reference angle

tan^{- 1} 1 = \frac{π}{4}

, and its principal argument is

- \frac{3 π}{4}

JEE Main · 2021 · 20 Jul 2021 · Shift 1 · Single correct option