Let `Z_(1)` and `Z_(2)` be two complex numbers satisfying `|Z_(1)|=9` and `|Z_(2)-3-4i|=4`. Then the minimum value of `|Z_(1)-Z_(2)|` is

Options

A. (a) 0
B. (b) 1
C. (c) `sqrt(2)`
D. (d) 2

Answer

To solve the problem, we need to find the minimum value of \( |Z_1 - Z_2| \) given the conditions on the complex numbers \( Z_1 \) and \( Z_2 \).

### Step-by-Step Solution:

1. **Understanding the Magnitudes**:
- We are given that \( |Z_1| = 9 \). This means that \( Z_1 \) lies on a circle centered at the origin (0, 0) with a radius of 9.
- We are also given that \( |Z_2 - (3 + 4i)| = 4 \). This means that \( Z_2 \) lies on a circle centered at the point (3, 4) with a radius of 4.

2. **Equations of the Circles**:
- The equation for the first circle (for \( Z_1 \)) can be expressed as:
\[ x_1^2 + y_1^2 = 9^2 = 81 \]
- The equation for the second circle (for \( Z_2 \)) can be expressed as:
\[ (x_2 - 3)^2 + (y_2 - 4)^2 = 4^2 = 16 \]

3. **Finding the Distance Between the Centers**:
- The center of the first circle \( C_1 \) is at (0, 0) and the center of the second circle \( C_2 \) is at (3, 4).
- The distance \( d \) between the centers \( C_1 \) and \( C_2 \) is calculated using the distance formula:
\[ d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

4. **Finding the Radii**:
- The radius of the first circle \( R_1 \) is 9.
- The radius of the second circle \( R_2 \) is 4.

5. **Analyzing the Relationship Between the Circles**:
- We need to find the minimum distance \( |Z_1 - Z_2| \). The minimum distance between two circles occurs when they are touching each other.
- The distance between the centers \( d = 5 \) and the difference in the radii \( |R_1 - R_2| = |9 - 4| = 5 \).
- Since \( d = |R_1 - R_2| \), the circles touch internally.

6. **Conclusion**:
- When two circles touch internally, the minimum distance between any point on the first circle and any point on the second circle is 0.
- Therefore, the minimum value of \( |Z_1 - Z_2| \) is:
\[ \boxed{0} \]