Key Concepts and Formulas

• Geometric Interpretation of $|z - z_0| \leq r$: This represents a closed disk in the complex plane centered at $z_0$ with radius $r$.
• Distance between complex numbers: The distance between $z_1$ and $z_2$ is given by $|z_1 - z_2|$.
• Minimum distance between two disks: If the distance between the centers of two disks is $d$, and their radii are $r_1$ and $r_2$, then the minimum distance between any two points in the disks is $d - r_1 - r_2$ if $d > r_1 + r_2$ (disks do not overlap), and 0 if $d \leq r_1 + r_2$ (disks overlap or touch).

Step-by-Step Solution

1. Identify the Centers and Radii of the Disks

We are given $|z_1 - 8 - 2i| \leq 1$ and $|z_2 - 2 + 6i| \leq 2$. We want to rewrite these in the form $|z - z_0| \leq r$ to identify the center $z_0$ and radius $r$ for each disk.

• For $z_1$, we have $|z_1 - (8 + 2i)| \leq 1$. The center is $C_1 = 8 + 2i$ and the radius is $r_1 = 1$.
• For $z_2$, we have $|z_2 - (2 - 6i)| \leq 2$. The center is $C_2 = 2 - 6i$ and the radius is $r_2 = 2$.

Explanation: We rewrite the given inequalities into the standard form to clearly identify the center and radius of each disk. The center is the complex number being subtracted from $z$ inside the absolute value.*

2. Calculate the Distance between the Centers

We need to find the distance $d$ between the centers $C_1 = 8 + 2i$ and $C_2 = 2 - 6i$. We use the formula $d = |C_1 - C_2|$.

$d = |(8 + 2i) - (2 - 6i)|$ $d = |8 + 2i - 2 + 6i|$ $d = |6 + 8i|$ $d = \sqrt{6^2 + 8^2}$ $d = \sqrt{36 + 64}$ $d = \sqrt{100}$ $d = 10$

Explanation: We calculate the direct distance between the two centers in the complex plane. This distance will be used to determine if the disks overlap and to calculate the minimum separation between any points within the two disks.*

3. Compare the Distance between Centers with the Sum of Radii and Calculate the Minimum Distance

We compare $d$ with $r_1 + r_2$. $d = 10$ and $r_1 + r_2 = 1 + 2 = 3$. Since $d > r_1 + r_2$, the disks do not overlap.

The minimum distance between $z_1$ and $z_2$ is given by: $|z_1 - z_2|_{\min} = d - r_1 - r_2$ $|z_1 - z_2|_{\min} = 10 - 1 - 2$ $|z_1 - z_2|_{\min} = 7$

Explanation: Since the disks do not overlap, the shortest distance between a point in the first disk and a point in the second disk occurs when both points lie on the line segment connecting the two centers. This minimum distance is found by taking the distance between centers and subtracting both radii.*

Common Mistakes & Tips

• Sign errors: Be careful with the signs when identifying the center of the disk from the inequality. Remember that $|z - z_0| \leq r$ means the center is $z_0$, not $-z_0$.
• Overlapping disks: Always check if the disks overlap ($d \leq r_1 + r_2$). If they do, the minimum distance is 0.
• Visualize: Sketching the disks in the complex plane can help with understanding the problem.

Summary

The minimum value of $|z_1 - z_2|$ is 7. We found this by identifying the centers and radii of the disks defined by the given inequalities, calculating the distance between the centers, and then subtracting the radii from the distance since the disks did not overlap.

Final Answer

The final answer is $\boxed{7}$, which corresponds to option (C).