For all complex numbers $z_1\text{ and }{z}_2$ satisfying 

$\left|z_1\right|=12\text{ and }\left|z_2-3-4i\right|=5$, the minimum value of

$\left|z_1-z_2\right|$ is

$\left|z_1\right|=12$ is circle with centre (0, 0) and radius 12. 

$\left|z_2-3-4i\right|=5$ is circle with centre (3, 4) and radius 5. 

The minimum distance between $z_1$

and $z_2\text{ in }AB=12-10=2$.