Let $|z_{1}-8-2i|\le 1$ and $|z_{2}-2+6i|\le 2$, $z_{1},z_{2}\in\mathbb{C}$. Then the minimum value of $|z_{1}-z_{2}|$ is:
Step-by-Step Solution
Key Concept: Two closed disks: minimum distance between any pair of points is $\max(0,\ d(C_1,C_2)-r_1-r_2)$.
Disks: $D_1$ centered at $C_1=8+2i$, radius $1$; $D_2$ centered at $C_2=2-6i$, radius $2$.
$$|C_1-C_2| = |6+8i| = \sqrt{36+64} = \sqrt{100}=10.$$
Since $10 > 1+2=3$, the disks are disjoint. Minimum separation
$$\min|z_{1}-z_{2}| = 10-1-2 = 7.$$
Correct Answer: 4