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: Understanding the Concept:
The distance between two complex numbers \(z_1\) and \(z_2\) is geometrically represented by \(|z_1 - z_2|\).
For two circles where one lies completely inside the other, the minimum and maximum distances between points on the two circles occur along the line connecting their centers.
Step 2: Key Formula or Approach:
Let \(C_1\) have center \(O_1(0,0)\) and radius \(r\).
Let \(C_2\) have center \(O_2(3,4)\) and radius \(R = 5\).
The distance between their centers is \(d = |3 + 4i| = \sqrt{3^2 + 4^2} = 5\).
Since \(C_2\) lies completely inside \(C_1\), the minimum distance is \(r - (d + R)\) and the maximum distance is \(r + d + R\).
Step 3: Detailed Explanation:
Given that the minimum distance is 2, we can set up the equation for \(r\).
\[ \min |z_1 - z_2| = r - (d + R) = 2 \] Substitute the known values \(d = 5\) and \(R = 5\) into the equation.
\[ r - (5 + 5) = 2 \] \[ r - 10 = 2 \implies r = 12 \] Now we need to find the maximum possible distance between a point on \(C_1\) and a point on \(C_2\).
\[ \max |z_1 - z_2| = r + (d + R) \] \[ \max |z_1 - z_2| = 12 + (5 + 5) = 12 + 10 = 22 \] Step 4: Final Answer:
The maximum distance \(\max |z_1 - z_2|\) is \(22\).