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

$\left|z_1\right|=12$  implies that $z_1$  lies on the circle with center $C_1$  at origin and radius 12 whereas $|z_2-3-4i|=5$  implies $z_2$  lies on the circle with centre at $C_2(3+4i)$  and radius 5.

See The quantity $|z_1-z_2|$  will attain the least value 2 when $z_1$  and $z_2$  lie on the line joining $C_1$  and $C_2$  i.e. on the line $z=(3+4i)t$

In fact, when we take $z_2=6+8i$  and $z_1=\frac{12}{5}(3+4i),$  we obtain $|z_1-z_2|=\left|\frac{12}{5}(3+4i)-(6+8i)\right|=\frac{2}{5}|3+4i|=2.$