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 that \( z_1, z_2, z_3, z_4 \) are distinct complex numbers satisfying \( |z| = 1 \) and \( 4z_3 = 3(z_1 + z_2) \).

### Step-by-Step Solution:

1. **Understanding the Condition**:
Since \( |z| = 1 \), all the complex numbers \( z_1, z_2, z_3, z_4 \) lie on the unit circle in the complex plane. This means that the modulus of each of these complex numbers is 1.

2. **Rearranging the Given Equation**:
We start with the equation:
\[ 4z_3 = 3(z_1 + z_2) \]
Dividing both sides by 4 gives:
\[ z_3 = \frac{3}{4}(z_1 + z_2) \]

3. **Finding the Midpoint**:
We can express \( z_1 + z_2 \) in terms of \( z_3 \):
\[ z_1 + z_2 = \frac{4}{3}z_3 \]
This means that \( z_3 \) is the centroid of the points \( z_1 \) and \( z_2 \) when scaled appropriately.

4. **Using the Properties of the Circle**:
Since \( z_1 \) and \( z_2 \) are on the unit circle, we can denote them as:
\[ z_1 = e^{i\theta_1}, \quad z_2 = e^{i\theta_2} \]
where \( \theta_1 \) and \( \theta_2 \) are angles on the unit circle.

5. **Finding the Distance**:
The distance \( |z_1 - z_2| \) can be computed using the formula:
\[ |z_1 - z_2| = |e^{i\theta_1} - e^{i\theta_2}| = |e^{i\theta_1}(1 - e^{i(\theta_2 - \theta_1)})| = |1 - e^{i(\theta_2 - \theta_1)}| \]
This simplifies to:
\[ |z_1 - z_2| = 2 \sin\left(\frac{\theta_2 - \theta_1}{2}\right) \]

6. **Using the Centroid Property**:
Since \( z_3 \) is the centroid, we can use the fact that the distance from the origin to the midpoint of \( z_1 \) and \( z_2 \) is given by:
\[ |z_3| = \frac{1}{2}|z_1 + z_2| = \frac{1}{2} \left| \frac{4}{3} z_3 \right| = \frac{2}{3} \]

7. **Applying the Pythagorean Theorem**:
The distance \( |z_1 - z_2| \) can also be computed using the Pythagorean theorem in the triangle formed by the origin and the points \( z_1 \) and \( z_2 \):
\[ |z_1 - z_2| = 2 \sqrt{1 - \left(\frac{2}{3}\right)^2} = 2 \sqrt{1 - \frac{4}{9}} = 2 \sqrt{\frac{5}{9}} = \frac{2\sqrt{5}}{3} \]

### Final Answer:
Thus, the value of \( |z_1 - z_2| \) is:
\[ \boxed{\frac{2\sqrt{5}}{3}} \]