QJEEM21Ma44d3: JEE Main 2021 Mathematics - Class 11 JEEM21Ma44d3 Solution: If z and are two complex numbers such that and , then is :…

Question

Verified Solution • 2025–26 Session moderate

If z and

\omega

are two complex numbers such that

\left| {z\omega } \right| = 1

and

\arg (z) - \arg (\omega ) = {{3\pi } \over 2}

, then

\arg \left( {{{1 - 2\overline z \omega } \over {1 + 3\overline z \omega }}} \right)

is :

(Here arg(z) denotes the principal argument of complex number z)

Accepted Answer

As

\left| {z\omega } \right| = 1

\Rightarrow

If

\left| z \right| = r

, then

\left| \omega \right| = {1 \over r}

Let

\arg (z) = \theta

\therefore

\arg (\omega ) = \left( {\theta - {{3\pi } \over 2}} \right)

So,

z = r{e^{i\theta }}

\Rightarrow \overline z = r{e^{i\theta }}

\omega = {1 \over r}{e^{i\left( {\theta - {{3\pi } \over 2}} \right)}}

Now, consider

{{1 - 2\overline z \omega } \over {1 + 3\overline z \omega }} = {{1 - 2{e^{i\left( { - {{3\pi } \over 2}} \right)}}} \over {1 + 3{e^{i\left( { - {{3\pi } \over 2}} \right)}}}} = \left( {{{1 - 2i} \over {1 + 3i}}} \right)

= {{(1 - 2i)(1 - 3i)} \over {(1 + 3i)(1 - 3i)}} = - {1 \over 2}(1 + i)

\therefore

prin\arg \left( {{{1 - 2\overline z \omega } \over {1 + 3\overline z \omega }}} \right)

= prin\arg \left( {{{1 - 2\overline z \omega } \over {1 + 3\overline z \omega }}} \right)

= \left( { - {1 \over 2}(1 + i)} \right)

= - \left( {\pi - {\pi \over 4}} \right) = {{ - 3\pi } \over 4}

So, option (2) is correct.

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