JEE MAIN - Mathematics (2002 - No. 36)
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
$$\overline \omega $$
$$ - \overline \omega $$
$$\omega $$
$$ - \omega $$
Explanation
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 $$\,\,\,\,\overline \omega = r{e^{ - i\phi }}$$ ]
$$\therefore$$ $$z = r{e^{i\theta }},\omega = r{e^{i\phi }}$$
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 $$\,\,\,\,\overline \omega = r{e^{ - i\phi }}$$ ]
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