If `z_(1)` satisfies `|z-1|=1` and `z_(2)` satisfies `|z-4i|=1`, then `|z_(1)-z_(2)|_("max")-|z_(1)-z_(2)|_("min")` is equal to ___________

Answer

To solve the problem, we need to find the maximum and minimum values of the expression \(|z_1 - z_2|\) where \(z_1\) satisfies \(|z - 1| = 1\) and \(z_2\) satisfies \(|z - 4i| = 1\).

### Step 1: Determine the values of \(z_1\)

The equation \(|z - 1| = 1\) represents a circle in the complex plane centered at \(1\) (which is \(1 + 0i\)) with a radius of \(1\). The points on this circle can be expressed as:

\[ z_1 = 1 + e^{i\theta} \quad \text{for } \theta \in [0, 2\pi) \]

This gives us two specific points when we consider the extremes:

1. When \(\theta = 0\), \(z_1 = 1 + 1 = 2\).
2. When \(\theta = \pi\), \(z_1 = 1 - 1 = 0\).

Thus, the possible values for \(z_1\) are \(2\) and \(0\).

### Step 2: Determine the values of \(z_2\)

The equation \(|z - 4i| = 1\) represents another circle in the complex plane centered at \(4i\) (which is \(0 + 4i\)) with a radius of \(1\). The points on this circle can be expressed as:

\[ z_2 = 4i + e^{i\phi} \quad \text{for } \phi \in [0, 2\pi) \]

This gives us two specific points when we consider the extremes:

1. When \(\phi = 0\), \(z_2 = 0 + 4i + 1 = 1 + 4i\).
2. When \(\phi = \pi\), \(z_2 = 0 + 4i - 1 = -1 + 4i\).

Thus, the possible values for \(z_2\) are \(1 + 4i\) and \(-1 + 4i\).

### Step 3: Calculate \(|z_1 - z_2|\)

Now we need to calculate \(|z_1 - z_2|\) for the combinations of the values of \(z_1\) and \(z_2\).

1. **For \(z_1 = 2\) and \(z_2 = 1 + 4i\)**:
\[ |z_1 - z_2| = |2 - (1 + 4i)| = |2 - 1 - 4i| = |1 - 4i| = \sqrt{1^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17} \]

2. **For \(z_1 = 2\) and \(z_2 = -1 + 4i\)**:
\[ |z_1 - z_2| = |2 - (-1 + 4i)| = |2 + 1 - 4i| = |3 - 4i| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

3. **For \(z_1 = 0\) and \(z_2 = 1 + 4i\)**:
\[ |z_1 - z_2| = |0 - (1 + 4i)| = |-1 - 4i| = \sqrt{(-1)^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17} \]

4. **For \(z_1 = 0\) and \(z_2 = -1 + 4i\)**:
\[ |z_1 - z_2| = |0 - (-1 + 4i)| = |1 - 4i| = \sqrt{1^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17} \]

### Step 4: Find the maximum and minimum values

From the calculations, we have the following values for \(|z_1 - z_2|\):

- \(\sqrt{17}\)
- \(5\)
- \(\sqrt{17}\)
- \(\sqrt{17}\)

The maximum value is \(5\) and the minimum value is \(\sqrt{17}\).

### Step 5: Calculate the final result

Now, we need to find:

\[ |z_1 - z_2|_{\text{max}} - |z_1 - z_2|_{\text{min}} = 5 - \sqrt{17} \]

### Conclusion

Thus, the final answer is:

\[ 5 - \sqrt{17} \]