Let `z_(1),z_(2),z_(3),z_(4)` are distinct complex numbers satisfying `|z|=1` and `4z_(3) = 3(z_(1) + z_(2))`, then `|z_(1) - z_(2)|` is equal to

Options

A. 1 or i
B. `i or -i`
C. 1 or i
D. `i or -1`

Explanation

To solve the problem, we need to find the value of \( |z_1 - z_2| \) given the conditions on the complex numbers \( z_1, z_2, z_3, z_4 \).

### Step-by-Step Solution:

1. **Understanding the Given Conditions**:
We know that \( |z| = 1 \) for \( z_1, z_2, z_3, z_4 \). This means that these complex numbers lie on the unit circle in the complex plane.

2. **Using the Given Equation**:
We have the equation \( 4z_3 = 3(z_1 + z_2) \). We can rearrange this to express \( z_1 + z_2 \):
\[ z_1 + z_2 = \frac{4}{3} z_3 \]

3. **Finding the Midpoint**:
The midpoint \( M \) of \( z_1 \) and \( z_2 \) can be expressed as:
\[ M = \frac{z_1 + z_2}{2} = \frac{2}{3} z_3 \]

4. **Using the Section Formula**:
Since \( z_1 \) and \( z_2 \) are on the unit circle, we can use the section formula. The point \( z_3 \) divides the line segment joining \( z_1 \) and \( z_2 \) in the ratio \( 2:1 \). Thus, we can write:
\[ z_3 = \frac{2z_2 + z_1}{3} \]

5. **Finding the Length \( |z_1 - z_2| \)**:
To find \( |z_1 - z_2| \), we can use the fact that \( |z_1 - z_2| = 2|AM| \), where \( A \) and \( B \) are the points corresponding to \( z_1 \) and \( z_2 \), and \( M \) is the midpoint.

6. **Calculating \( |AM| \)**:
The length \( |AM| \) can be calculated using the formula:
\[ |AM| = \sqrt{|OA|^2 - |OM|^2} \]
where \( |OA| \) is the radius of the circle (which is 1) and \( |OM| \) is the distance from the origin to the midpoint \( M \).

7. **Finding \( |OM| \)**:
Since \( M = \frac{2}{3} z_3 \) and \( |z_3| = 1 \), we have:
\[ |OM| = \left| \frac{2}{3} z_3 \right| = \frac{2}{3} \]

8. **Final Calculation**:
Now substituting back:
\[ |AM| = \sqrt{1^2 - \left(\frac{2}{3}\right)^2} = \sqrt{1 - \frac{4}{9}} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3} \]
Therefore,
\[ |z_1 - z_2| = 2 |AM| = 2 \cdot \frac{\sqrt{5}}{3} = \frac{2\sqrt{5}}{3} \]

### Conclusion:
Thus, the value of \( |z_1 - z_2| \) is \( \frac{2\sqrt{5}}{3} \).