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 \( \boxed{7} \).

Answer 2

Step 1: Understand the given conditions.
We are given two complex numbers \( z_1 \) and \( z_2 \) that satisfy the following conditions:
1. \( |z_1 - 8 - 2i| \leq 1 \), which represents a disk with center \( (8, 2) \) and radius 1 in the complex plane.
2. \( |z_2 - 2 + 6i| \leq 2 \), which represents a disk with center \( (2, -6) \) and radius 2 in the complex plane.
The problem asks to find the minimum value of \( |z_1 - z_2| \), which represents the minimum distance between any point in the first disk and any point in the second disk.

Step 2: Visualize the problem.
We need to calculate the minimum distance between the two disks: - Disk 1 has center \( (8, 2) \) and radius 1. - Disk 2 has center \( (2, -6) \) and radius 2.
Step 3: Calculate the distance between the centers of the disks.
The distance between the centers of the two disks is given by the distance formula: \[ \text{Distance} = \sqrt{(8 - 2)^2 + (2 - (-6))^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10. \] Step 4: Calculate the minimum distance between the disks.
The minimum distance between the disks occurs when the line connecting the centers of the disks intersects both disks. The minimum distance is the distance between the centers minus the radii of the two disks: \[ \text{Minimum distance} = 10 - (1 + 2) = 10 - 3 = 7. \] Final Answer:
\[ \boxed{7}. \]