JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Mock Test - Complex Numbers and Quadratic Equations

If z and \[\omega \]are two non-zero complex numbers such that \[\left| z \right|=\left| \omega \right|\]and arg z + arg \[\omega \]=\[\pi \], then z equals

A) \[\bar{\omega }\]

B) -\[\bar{\omega }\]

C) \[\omega \]

D) -\[\omega \]

Correct Answer: B

Solution :

[b] Let\[\left| z \right|=\left| \omega \right|=r\]

\[\therefore z=r{{e}^{i\theta }},\omega =r{{e}^{i\theta }}\]

Where \[\theta +\phi =\pi \]

\[\therefore \overline{\omega }=r{{e}^{-i\phi }}\]

\[\therefore z=r{{e}^{i(\pi -\phi )}}=r{{e}^{i\pi }}{{e}^{-i\phi }}=r{{-e}^{-i\phi }}=-\overline{\omega }\]