Question
$$z$$ and $$w$$ are two nonzero complex numbers such that $$\left| z \right| = \left| w \right|\,\,{\text{and Arg}}\,z + {\text{Agr}}\,w = \pi $$ then $$z$$ equals
A.
$$\overline \omega $$
B.
$$ - \overline \omega $$
C.
$$\omega $$
D.
$$ - \omega $$
Answer :
$$ - \overline \omega $$
Solution :
$$\eqalign{
& z = \left| z \right|\left( {\cos \,\theta + i\,\sin \,\theta } \right) \cr
& {\text{where }}\theta = {\text{Arg }}z \cr
& {\text{if }}{\theta _1} = {\text{Arg }}w{\text{ then }}\theta = \pi - {\theta _1} \cr
& \therefore \,z = \left| w \right|\left\{ {\cos \left( {\pi - {\theta _1}} \right) + i\,\sin \left( {\pi - {\theta _1}} \right)} \right\} \cr
& \Rightarrow z = \left| w \right|\left( { - \cos \,{\theta _1} + i\,\sin \,{\theta _1}} \right) \cr
& \Rightarrow z = - \left| w \right|\left( { - \cos \,{\theta _1} + i\,\sin \,{\theta _1}} \right) \cr
& \Rightarrow z = - \left| w \right|\left( {\cos \,{\theta _1} - i\,\sin \,{\theta _1}} \right) \cr
& \Rightarrow z = - \left| {\overline w } \right| \cr} $$