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. 3
• C. 10
• D. 7

Hint:

The minimum distance between two points on two circles is the distance between their centers minus the sum of their radii.

Answer 1

The Correct Option is D

We are tasked with finding the minimum distance between two circles, \( C_1 \) and \( C_2 \), given their centers and radii. Let us proceed step by step:

1. Given Information:
The centers and radii of the circles are:

\( C_1(8, 2), \quad r_1 = 1 \)
\( C_2(2, 6), \quad r_2 = 2 \)

2. Distance Between the Centers:
The distance between the centers \( C_1 \) and \( C_2 \) is given by the Euclidean distance formula:

\( C_1C_2 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
 

3. Minimum Distance Between the Circles:
The minimum distance between the two circles is the distance between their centers minus the sum of their radii:

\( |z_1 - z_2| = C_1C_2 - (r_1 + r_2) \)

Substitute \( C_1C_2 = 10 \), \( r_1 = 1 \), and \( r_2 = 2 \):

\( |z_1 - z_2| = 10 - (1 + 2) \)

\( |z_1 - z_2| = 10 - 3 = 7 \)

Final Answer:
The minimum distance between the two circles is \( \boxed{7} \).