Question

Let the circles \( C_1 : |z| = r \) and \( C_2 : |z - 3 - 4i| = 5, z \in \mathbb{C} \), be such that \( C_2 \) lies within \( C_1 \). If \( z_1 \) moves on \( C_1 \), \( z_2 \) moves on \( C_2 \) and \( \min |z_1 - z_2| = 2 \), then \( \max |z_1 - z_2| \) is equal to:

• A. 12
• B. 17
• C. 22
• D. 24

Answer 1

The Correct Option is A

Step 1: Understand the given conditions.
We are given two circles:
- \( C_1 \): Center at the origin \( O_1 = 0 \), radius \( r \).
- \( C_2 \): Center at \( O_2 = 3 + 4i \), radius \( 5 \).
The minimum distance between a point on \( C_1 \) and a point on \( C_2 \) is \( 2 \), and we are asked to find the maximum possible distance between the points.
Step 2: Calculate the distance between the centers of the circles.
The distance between the centers \( O_1 \) and \( O_2 \) is: \[ d = |O_1 - O_2| = |0 - (3 + 4i)| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Step 3: Maximum distance calculation.
The maximum distance between a point on \( C_1 \) and a point on \( C_2 \) occurs when the points are on the line connecting the centers of the two circles, and the points are at the farthest ends of their respective circles. The maximum distance is: \[ \text{Maximum distance} = d + r_1 + r_2 = 5 + 5 + 2 = 12 \]
Final Answer: 12