Question

If \( z_1 \) lies on curve \( |z| = r \) and \( z_2 \) lies on curve \( |z - 3 - 4i| = 5 \), if minimum of \( |z_1 - z_2| = 2 \), then the maximum of \( |z_1 - z_2| \) is

• A. 12
• B. 18
• C. 20
• D. 22

Hint:

For two circles, minimum distance between points is usually the difference between the distance of centers and the sum or difference of radii depending on their position, while maximum distance is the distance between centers plus both radii.

Answer 1

The Correct Option is D

Step 1: Interpret the given loci geometrically.
The equation \( |z_1| = r \) represents a circle centered at the origin \( O(0,0) \) with radius \( r \). The equation \( |z_2 - (3 + 4i)| = 5 \) represents another circle centered at the point \( C(3,4) \) with radius \( 5 \). The distance between the centers of the two circles is: \[ OC = \sqrt{3^2 + 4^2} = 5. \]
Step 2: Use the given minimum distance condition.
The minimum distance between two points on the circumferences of the circles is given as \( 2 \). For two circles, the minimum distance between their circumferences (when they do not intersect) is: \[ \text{Minimum distance} = |\,OC - (r + 5)\,|. \] Substituting the known values: \[ |\,5 - (r + 5)\,| = 2. \] \[ |\,r\,| = 2. \] Since radius is always positive: \[ r = 2. \]
Step 3: Determine the maximum distance between the two circles.
The maximum distance between points on the two circles occurs when the points lie along the line joining their centers in opposite directions. Thus, the maximum distance is: \[ \text{Maximum distance} = OC + r + 5. \] Substituting the values: \[ = 5 + 2 + 5 = 12. \]
Step 4: Conclusion.
Therefore, using the geometric interpretation of the loci and the minimum distance condition, the maximum possible value of \( |z_1 - z_2| \) is:
Final Answer: \( 12 \).