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. 1
B. 2
C. `sqrt(2)`
D. 0

Explanation

To solve the problem, we need to find the minimum value of \( |z_1 - z_2| \) given the conditions \( |z_1| = 9 \) and \( |z_2 - (3 + 4i)| = 4 \).

### Step 1: Understand the Magnitudes
The condition \( |z_1| = 9 \) means that \( z_1 \) lies on a circle of radius 9 centered at the origin in the complex plane.

The condition \( |z_2 - (3 + 4i)| = 4 \) means that \( z_2 \) lies on a circle of radius 4 centered at the point \( (3, 4) \) in the complex plane.

### Step 2: Represent the Circles
1. The circle for \( z_1 \) can be represented as:
\[ z_1 = 9e^{i\theta} \quad \text{for } \theta \in [0, 2\pi) \]
This represents all points on the circle of radius 9.

2. The circle for \( z_2 \) can be represented as:
\[ z_2 = 3 + 4i + 4e^{i\phi} \quad \text{for } \phi \in [0, 2\pi) \]
This represents all points on the circle of radius 4 centered at \( (3, 4) \).

### Step 3: Find the Distance Between the Centers
Next, we find the distance between the center of the circle for \( z_1 \) (which is at the origin \( (0, 0) \)) and the center of the circle for \( z_2 \) (which is at \( (3, 4) \)):
\[ d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Calculate the Minimum Distance
To find the minimum distance between any point on the circle of radius 9 (for \( z_1 \)) and any point on the circle of radius 4 (for \( z_2 \)), we can use the following relationship:
\[ \text{Minimum distance} = d - r_1 - r_2 \]
where \( r_1 = 9 \) (radius of \( z_1 \)) and \( r_2 = 4 \) (radius of \( z_2 \)).

Substituting the values:
\[ \text{Minimum distance} = 5 - 9 - 4 = 5 - 13 = -8 \]

Since distance cannot be negative, this indicates that the circles overlap. Therefore, the minimum distance is actually 0.

### Conclusion
The minimum value of \( |z_1 - z_2| \) is:
\[ \boxed{0} \]