Question
Let $$z$$ and $$\omega $$ be two non zero complex numbers such that $$\left| z \right| = \left| \omega \right|\,\,{\text{and Arg }}z + {\text{Arg }}\omega = \pi ,$$ then $$z$$ equals
A.
$$\omega $$
B.
$$ - \omega $$
C.
$$ \overline \omega $$
D.
$$ - \overline \omega $$
Answer :
$$ - \overline \omega $$
Solution :
$$\eqalign{
& \because \,\,\left| z \right| = \left| \omega \right|\,\,{\text{and arg }}z = \pi - \,{\text{arg }}\omega \cr
& {\text{Let }}\omega = r{e^{i\theta }}\,\,{\text{then }}z = r{e^{i\left( {\pi - \theta } \right)}} \cr
& \Rightarrow \,\,z = r{e^{i\pi }}.{e^{ - i\theta }} \cr
& = \left( {r{e^{ - i\theta }}} \right)\left( {\cos \pi + i\sin \pi } \right) = \overline \omega \left( { - 1} \right) = - \overline \omega \cr} $$