Question

If $z$ is a complex number such that $-\pi /2\le$ arg $z\le \pi /2,$ then which of the following inequality is true?

• A. $|z-\overline{z}|\le \left|z\right|$(arg $z$- arg $\overline{z})$
• B. $|z-\overline{z}|\ge \left|z\right|$(arg $z$ - arg $\overline{z})$
• C. $|z-\overline{z}|<\left|z\right|$(arg $z$ - arg $\overline{z})$
• D. none of these

Answer 1

The correct option is A $|z-\overline{z}|\le \left|z\right|$(arg $z$- arg $\overline{z})$

Let $z=\left|z\right|(\cos A+i\sin A)whereA=arg\left(z\right),-\frac{\Pi }{2}\le A\le \frac{\Pi }{2}$
$|z-\overline{z}|=\left|z\right||\cos A+i\sin A-(\cos A-i\sin A)|=2\left|z\right|\left|i\sin A\right|$
$⟹|z-\overline{z}|=2\left|z\right|\left|\sin A\right|$
$for-\frac{\pi }{2}\le A\le \frac{\pi }{2}\left|\sin A\right|\le |A|$
$⟹|z-\overline{z}|\le 2\left|z\right|\left|A\right|=\left|z\right||A-(-A)|$
$⟹|z-\overline{z}|\le \left|z\right||argz-arg\overline{z}|$
Ans: A