For all complex numbers $z_1,z_2$  satisfying $\left|z_1\right|=12$  and $\left|z_2-3-4i\right|=5$ , the minimum value of $\left|z_1-z_2\right|$  is

$\left|z_1\right|=12\Rightarrow z_1$  lies on circle with centre $c_1$  at origin and radius 12

                $\left|z_2-3-4i\right|=5\Rightarrow z_2$  lies on the circle with centre $c_2$ $\left(3+4i\right)$  and radius 5.

$\therefore \left|z_1-z_2\right|$  will be minimum.

If $z_1$  and $z_2$  lies on the line joining $c_1$  and $c_2$  i.e on the line $z=3+4i$

Minimum value of $\left|z_1-z_2\right|=AB$

$=c_{1}B-c_{1}A$

=$12-10=2$