z and w are two nonzero complex numbers such that [,| z | = | w |] and Arg z + Arg w =[ ] …

Question

z and w are two nonzero complex numbers such that $$\,\left| z \right| = \left| w \right|$$ and Arg z + Arg w =$$\pi $$ then z equals

Accepted Answer

Let $$\left| z \right| = \left| \omega  \right| = r$$

$$\therefore$$ $$z = r{e^{i\theta }},\omega  = r{e^{i\phi }}$$

where $$\,\,\theta  + \phi  = \pi .$$

$$\therefore$$ $$z = r{e^{i\left( {\pi  - \phi } \right)}} = r{e^{i\pi }}.$$
$${e^{ - i\phi }} =  - r{e^{ - i\phi }} =  - \overline \omega  .$$

[as $$\,\,\,\,\overline \omega   = r{e^{ - i\phi }}$$ ]

Solutions · 1