Interpret the given loci geometrically

Let $|z_1-(8+2i)|\le 1$ and $|z_2-(2-6i)|\le 2.$

These represent two closed discs in the Argand plane:

• Disc $D_1$ with center $C_1=(8,2), \quad r_1=1$
• Disc $D_2$ with center $C_2=(2,-6), \quad r_2=2$

Find the distance between the centers

$C_1C_2=\sqrt{(8-2)^2+(2-(-6))^2}$ $=\sqrt{6^2+8^2}$ $=\sqrt{36+64}=\sqrt{100}=10.$

Minimum distance between two points from the discs

The minimum possible value of $|z_1-z_2|$ is the minimum distance between the two discs.

Since the discs are externally separate, $\min |z_1-z_2|=C_1C_2-(r_1+r_2).$

Therefore, $\min |z_1-z_2|=10-(1+2)=7.$

Check with options

The correct option is: $\boxed{7}$

So, Option C is correct.