Let $\left|{\mathrm{z}}_1-1\right|=1, \left|{\mathrm{z}}_2+4\right|=2$ 2then maximum value of $\left|{\mathrm{z}}_1-{\mathrm{z}}_2\right|$ is

$\text{ Maximum distance between two curves lies along their common normal. }$

${\mathrm{z}}_1\mathrm{lies} \mathrm{on} \mathrm{a} \mathrm{circle} \mathrm{with} \mathrm{centre} (1,0) \mathrm{and} \mathrm{radius} =1$

${\mathrm{z}}_2 \mathrm{lies} \mathrm{on} \mathrm{a} \mathrm{circle} \mathrm{with} \mathrm{cnetre}\left(-4,0 \right) \mathrm{and} \mathrm{radius} =2$

Required  value  is distance  between (-6,0) and (2,0)=8