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 define two disks in the complex plane. Disk 1 is centered at \( (8, 2) \) with a radius of 1. Disk 2 is centered at \( (2, -6) \) with a radius of 2.
Step 2: The minimum distance between two disks is found by subtracting the sum of their radii from the distance between their centers. First, calculate the distance between the centers \( (8, 2) \) and \( (2, -6) \): \[ d = \sqrt{(8 - 2)^2 + (2 - (-6))^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \] The minimum distance between the disks is therefore \( d - (1 + 2) = 10 - 3 = 7 \). The correct answer is (4).