<s>[INST] Max. and Min. value of $|z|$ in $\left|z+\frac{2}{z}\right| = 2\,$ If $z$ is a complex no. such that $\displaystyle \left|z+\frac{2}{z}\right| = 2\,$ Then find max. and min. value of $\left|z\right|$.
$\bf{My\; Try:}$ Given $\displaystyle \left|z+\frac{2}{z}\right| = 2\Rightarrow  \left|z+\frac{2}{z}\right|^2 = 2^2=4$.
So $\displaystyle \left(z+\frac{2}{z}\right)\cdot \left(\bar{z}+\frac{2}{\bar{z}}\right) = 4\Rightarrow \left|z\right|^2+2\left(\frac{z}{\bar{z}}+\frac{\bar{z}}{z}\right)+\frac{1}{|z|^2} = 4$.
Now how can I find the max. and min. values of $|z|$?
Help me please.
Thanks
 [/INST] Let $z=re^{i\theta}$, then your last step can be written as 
$$r^2+2\cos (2\theta)+\frac{1}{r^2}=4.$$
From this we can get
\begin{align*}
\left(r+\frac{1}{r}\right)^2 & = 6-2\cos (2\theta)
\end{align*}
Thus $2 \leq \left(r+\frac{1}{r}\right) \leq 2\sqrt{2}.$ Now try to find the range of $r$.
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