If $\left|\mathrm{z} - 3 + 2\mathrm{i}\right| \le 4$ then the difference between the greatest value and the least value of $\left|\mathrm{z}\right|$ is

$$\begin{gathered} \left|\mathrm{z}-3+2\mathrm{i}\right|\le 4 \\ \left|\mathrm{z}-(3-2\mathrm{i})\right|\le 4 \end{gathered}$$

represents a circle with centre C (3, -2) and radious 4. LEty P(0, 0) shortest distance=$\left|\mathrm{CP}-\mathrm{r}\right|=4-\sqrt{13}$

longest distance $=\left|\mathrm{CP}+\mathrm{r}\right|=+\sqrt{13}$ 

$$\begin{gathered} {\left|\mathrm{z}\right|}_{\mathrm{Max}}-{\left|\mathrm{z}\right|}_{\mathrm{Min}}=(4+\sqrt{13})-(4-\sqrt{13}) \\ =2\sqrt{13} \end{gathered}$$