Question

$|Z-\frac{2}{Z}|=1$ then find the maximum and minimum value of $\left|z\right|$

Answer 1

$|z-\frac{2}{z}|=1$
$|\left|z\right|-\left|\frac{2}{z}\right||\le |z-\frac{2}{z}|\le \left|z\right|+|-\frac{2}{z}|$
Let $\left|z\right|=r$
$\Rightarrow$ $|r-\frac{2}{r}|\le 1\le r+\frac{2}{r}$
$r+\frac{2}{r}\ge 1$
$\Rightarrow$ $rϵR$ (i)
and $|r-\frac{2}{r}|\le 1$
$\Rightarrow$ $-1\le r-\frac{2}{r}\le 1$
$\Rightarrow$ $rϵ[1,2]$ (ii)
$\therefore$ from (i) and (ii) $rϵ[1,2]$
$rϵ[1,2]$
${\left|z\right|}_{max}=2,{\left|z\right|}_{min}=1$