If `|z_1-z_0|=z_2-z_1=pi/2` , then find `z_0`.

Explanation

To solve the problem, we need to find \( z_0 \) given the conditions \( |z_1 - z_0| = \frac{\pi}{2} \) and \( z_2 - z_1 = \frac{\pi}{2} \).

### Step-by-Step Solution:

1. **Understanding the Conditions**:
We have two conditions:
\[ |z_1 - z_0| = \frac{\pi}{2} \]
\[ z_2 - z_1 = \frac{\pi}{2} \]
The first condition tells us that the distance between \( z_1 \) and \( z_0 \) is \( \frac{\pi}{2} \). The second condition tells us that \( z_2 \) is \( \frac{\pi}{2} \) units away from \( z_1 \).

2. **Expressing \( z_2 \)**:
From the second condition, we can express \( z_2 \) in terms of \( z_1 \):
\[ z_2 = z_1 + \frac{\pi}{2} \]

3. **Using the First Condition**:
Now, substituting \( z_2 \) into the first condition:
\[ |z_1 - z_0| = \frac{\pi}{2} \]
This implies that \( z_0 \) lies on a circle of radius \( \frac{\pi}{2} \) centered at \( z_1 \).

4. **Expressing \( z_0 \)**:
We can express \( z_0 \) in terms of \( z_1 \) using the polar form:
\[ z_0 = z_1 + \frac{\pi}{2} e^{i\theta} \]
where \( \theta \) is the angle corresponding to the direction from \( z_1 \) to \( z_0 \).

5. **Finding \( z_0 \)**:
Since we know \( z_2 = z_1 + \frac{\pi}{2} \), we can also express \( z_0 \) in terms of \( z_2 \):
\[ z_0 = z_2 - \frac{\pi}{2} e^{-i\theta} \]
Here, \( e^{-i\theta} \) indicates the opposite direction from \( z_2 \) back to \( z_0 \).

6. **Final Expression**:
To find a specific form of \( z_0 \), we can set \( \theta = \frac{\pi}{2} \) (which corresponds to a 90-degree rotation). Thus:
\[ z_0 = z_1 + \frac{\pi}{2} i \]

### Conclusion:
The value of \( z_0 \) can be expressed as:
\[ z_0 = z_1 + \frac{\pi}{2} i \]