If `z_(1) , z_(2)` and `z_(3) , z_(4)` are two pairs of conjugate complex numbers, then arg (z1/z4) +arg (z2/z3) . is equal to

Options

A. A. 0
B. B. `(pi)/(2)`
C. C. `(3pi)/(2)`
D. D. `pi`

Answer

To solve the problem, we need to find the value of \( \text{arg} \left( \frac{z_1}{z_4} \right) + \text{arg} \left( \frac{z_2}{z_3} \right) \).

### Step-by-Step Solution:

1. **Identify the Conjugate Relationships**:
Given that \( z_2 \) is the conjugate of \( z_1 \) and \( z_4 \) is the conjugate of \( z_3 \), we can express:
\[ z_2 = \overline{z_1}, \quad z_4 = \overline{z_3} \]

2. **Define Arguments**:
Let \( \text{arg}(z_1) = \theta_1 \) and \( \text{arg}(z_3) = \theta_2 \). Then, the arguments of the conjugates are:
\[ \text{arg}(z_2) = -\theta_1, \quad \text{arg}(z_4) = -\theta_2 \]

3. **Apply the Argument Property**:
We use the property of arguments:
\[ \text{arg} \left( \frac{a}{b} \right) = \text{arg}(a) - \text{arg}(b) \]
Thus, we can write:
\[ \text{arg} \left( \frac{z_1}{z_4} \right) = \text{arg}(z_1) - \text{arg}(z_4) = \theta_1 - (-\theta_2) = \theta_1 + \theta_2 \]
and
\[ \text{arg} \left( \frac{z_2}{z_3} \right) = \text{arg}(z_2) - \text{arg}(z_3) = -\theta_1 - \theta_2 \]

4. **Combine the Arguments**:
Now, we can combine the two results:
\[ \text{arg} \left( \frac{z_1}{z_4} \right) + \text{arg} \left( \frac{z_2}{z_3} \right) = (\theta_1 + \theta_2) + (-\theta_1 - \theta_2) \]

5. **Simplify the Expression**:
Simplifying the expression gives:
\[ \theta_1 + \theta_2 - \theta_1 - \theta_2 = 0 \]

### Conclusion:
Thus, we find that:
\[ \text{arg} \left( \frac{z_1}{z_4} \right) + \text{arg} \left( \frac{z_2}{z_3} \right) = 0 \]

### Final Answer:
The answer is \( 0 \).