`(z_(1),z_(2)) and (z_(3) , z_(4)) ` are two pairs of conjugate complex numbers then arg `(z_(1))/( z_(3)) + arg"" (z_(2))/( z_(4))` is equal to

Options

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

Answer

To solve the problem, we need to find the value of \( \arg\left(\frac{z_1}{z_3}\right) + \arg\left(\frac{z_2}{z_4}\right) \), given that \( z_1, z_2 \) are conjugate pairs and \( z_3, z_4 \) are also conjugate pairs.

### Step-by-Step Solution:

1. **Identify the Conjugate Relationships**:
- Since \( z_1 \) and \( z_2 \) are conjugate pairs, we can express \( z_2 \) as:
\[ z_2 = \overline{z_1} \]
- Similarly, since \( z_3 \) and \( z_4 \) are conjugate pairs, we can express \( z_4 \) as:
\[ z_4 = \overline{z_3} \]

2. **Express the Arguments**:
- We need to find:
\[ \arg\left(\frac{z_1}{z_3}\right) + \arg\left(\frac{z_2}{z_4}\right) \]
- Using the properties of arguments, we can combine the arguments:
\[ \arg\left(\frac{z_1}{z_3}\right) + \arg\left(\frac{z_2}{z_4}\right) = \arg\left(\frac{z_1}{z_3} \cdot \frac{z_2}{z_4}\right) \]

3. **Substitute the Conjugate Expressions**:
- Substitute \( z_2 \) and \( z_4 \) with their conjugate forms:
\[ \frac{z_1}{z_3} \cdot \frac{z_2}{z_4} = \frac{z_1}{z_3} \cdot \frac{\overline{z_1}}{\overline{z_3}} = \frac{z_1 \cdot \overline{z_1}}{z_3 \cdot \overline{z_3}} \]

4. **Simplify the Expression**:
- The numerator \( z_1 \cdot \overline{z_1} \) is equal to \( |z_1|^2 \) (the square of the modulus of \( z_1 \)).
- The denominator \( z_3 \cdot \overline{z_3} \) is equal to \( |z_3|^2 \) (the square of the modulus of \( z_3 \)).
- Therefore, we have:
\[ \frac{z_1 \cdot \overline{z_1}}{z_3 \cdot \overline{z_3}} = \frac{|z_1|^2}{|z_3|^2} \]

5. **Determine the Argument**:
- Since \( |z_1|^2 \) and \( |z_3|^2 \) are both positive real numbers, the argument of a positive real number is:
\[ \arg\left(\frac{|z_1|^2}{|z_3|^2}\right) = 0 \]

6. **Final Result**:
- Therefore, we conclude that:
\[ \arg\left(\frac{z_1}{z_3}\right) + \arg\left(\frac{z_2}{z_4}\right) = 0 \]

### Conclusion:
The final answer is:
\[ \arg\left(\frac{z_1}{z_3}\right) + \arg\left(\frac{z_2}{z_4}\right) = 0 \]