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#### Pranav

##### Well-known member

- Nov 4, 2013

- 428

**Problem:**

If $z$ is a complex number such that

$$\arg(z(1+\overline{z}))+\arg\left(\frac{|z|^2}{z-|z|^2}\right)=0$$

then

A)$\arg(\overline{z})=-\pi/2$

B)$\arg(z)=\pi/4$

C)$|\overline{z}|<1$

D)$\ln\left(\frac{1}{|z|}\right)\in (-\infty,\infty)$

**Attempt:**

From the fact that $|z|=z\overline{z}$, I simplified the given equation to the following:

$$\arg\left(\frac{1+\overline{z}}{1-\overline{z}}\right)=0$$

If the argument of a complex number is zero, then it is equal to its conjugate, hence

$$\frac{1+\overline{z}}{1-\overline{z}}=\frac{1+z}{1-z}$$

Solving gives me $z=\overline{z}$. What to do with this?

Any help is appreciated. Thanks!