If $\left|\mathrm{Z} - 2 + 2\mathrm{i}\right| = 1$, then the least and the greatest values of |Z| are respectively

$|\mathrm{z}-(2-2\mathrm{i}\left)\right|=1$ represents a circle with centre $C\left(2,-2\right)$ and radius  $\mathrm{r} = 1$

Let $\mathrm{P}\left(0, 0\right)$ min of $\left|\mathrm{z}\right| =$ Distance OA

$$\begin{gathered} = OC -\mathrm{r} \\ = \sqrt{8} - 1 = 2\sqrt{2} - 1 \\ \max \mathrm{of} \left|\mathrm{z}\right| = OB = OP + \mathrm{r} = \sqrt{8} + 1 \end{gathered}$$