Question

Let \( |z_1 - 8 - 2i| \leq 1 \) and \( |z_2 - 2 + 6i| \leq 2 \), where \( z_1, z_2 \in \mathbb{C} \). Then the minimum value of \( |z_1 - z_2| \) is:

• A. 13
• B. 7
• C. 10
• D. 3

Hint:

For minimum distance problems between circles in the complex plane, use the distance between their centers and subtract the sum of their radii to find the minimum distance.

Answer 1

The Correct Option is D

Step 1: The given inequalities describe two disks in the complex plane: one centered at \( (8, 2) \) with radius 1, and the other centered at \( (2, -6) \) with radius 2. 
Step 2: The minimum distance between two points in the complex plane is the distance between their centers minus the sum of their radii. We calculate the distance between the centers \( (8, 2) \) and \( (2, -6) \) first: \[ d = \sqrt{(8 - 2)^2 + (2 - (-6))^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \] Thus, the minimum distance between the two disks is \( d - (1 + 2) = 10 - 3 = 7 \). Thus, the correct answer is (4).