If $\mathrm{a}=3+4\mathrm{i}, {\mathrm{Z}}_1$ and ${\mathrm{Z}}_2$ are two complex numbers such that $\left|{\mathrm{Z}}_1\right|=3$ and $\left|{\mathrm{Z}}_2-\mathrm{a}\right|=2$ then the maximum value of $\left|{\mathrm{Z}}_1-{\mathrm{Z}}_2\right|$ is _

$\left|{\mathrm{Z}}_1\right|=3\Rightarrow Z_1 \mathrm{lies} \mathrm{on} \mathrm{a} \mathrm{circle} \mathrm{with} {\mathrm{centreC}}_1 (0,0) \mathrm{and} \mathrm{radius} {\mathrm{r}}_1=3$

$$\begin{gathered} \left|{\mathrm{Z}}_2-\left(3+4\mathrm{i}\right)\right|=2 \\ \Rightarrow Z_2 \mathrm{lies} \mathrm{on} \mathrm{a} \mathrm{circle} \mathrm{with} \mathrm{centre} {\mathrm{C}}_2=\left(3,4\right) \mathrm{and} \mathrm{radius} =2 \end{gathered}$$

$C_1{C}_2=5,r_1+r_2=5\Rightarrow C_1{C}_2=r_1+r_2$

given two circles touch  each other externally

$\mathrm{Maximum} \mathrm{value} \mathrm{of} \left|{\mathrm{Z}}_1-{\mathrm{Z}}_2\right|=2\left({\mathrm{r}}_1+{\mathrm{r}}_2\right)=10$