Let `z _(1) and z _(2)` be two complex numberjs satisfying `|z _(1)|=3 and |z _(2) -3-4i|=4.` Then the minimum value of `|z_(1) -z _(2)|` is

Answer

To find the minimum value of \( |z_1 - z_2| \) given the conditions \( |z_1| = 3 \) and \( |z_2 - 3 - 4i| = 4 \), we can follow these steps:

### Step 1: Understand the conditions
The condition \( |z_1| = 3 \) means that the complex number \( z_1 \) lies on a circle centered at the origin (0, 0) with a radius of 3.

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

### Step 2: Identify the centers and radii of the circles
- The center of the circle for \( z_1 \) is at (0, 0) with a radius of 3.
- The center of the circle for \( z_2 \) is at (3, 4) with a radius of 4.

### Step 3: Calculate the distance between the centers of the circles
To find the minimum distance between any point on the circle of \( z_1 \) and any point on the circle of \( z_2 \), we first calculate the distance between the centers of the two circles:
\[ d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5. \]

### Step 4: Determine the minimum distance between the two circles
The minimum distance between the two circles can be found by subtracting the radii of the circles from the distance between their centers:
- The radius of the circle for \( z_1 \) is 3.
- The radius of the circle for \( z_2 \) is 4.

Thus, the minimum distance \( d_{min} \) is:
\[ d_{min} = d - (r_1 + r_2) = 5 - (3 + 4) = 5 - 7 = -2. \]
Since the distance cannot be negative, this indicates that the circles overlap.

### Step 5: Conclusion about the minimum value of \( |z_1 - z_2| \)
Since the circles overlap, the minimum value of \( |z_1 - z_2| \) is 0, which occurs when \( z_1 \) and \( z_2 \) coincide at some point.

Thus, the minimum value of \( |z_1 - z_2| \) is:
\[ \boxed{0}. \]