If `Z_1,Z_2` and `Z_3 , Z_4` are two pairs of conjugate complex numbers then `arg (Z_1/Z_4) + arg (Z_2/Z_3)` equals:

Options

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

Explanation

To solve the problem, we need to analyze the given pairs of conjugate complex numbers and use properties of complex numbers, particularly the argument of a quotient.

### Step-by-Step Solution:

1. **Understanding Conjugate Complex Numbers**:
- Let \( Z_1 = a + bi \) and \( Z_2 = a - bi \) (first pair of conjugates).
- Let \( Z_3 = c + di \) and \( Z_4 = c - di \) (second pair of conjugates).

2. **Finding the Quotients**:
- We need to calculate \( \frac{Z_1}{Z_4} \) and \( \frac{Z_2}{Z_3} \).
- The first quotient is:
\[ \frac{Z_1}{Z_4} = \frac{a + bi}{c - di} \]
- The second quotient is:
\[ \frac{Z_2}{Z_3} = \frac{a - bi}{c + di} \]

3. **Using the Argument Property**:
- The property of arguments states that:
\[ \arg\left(\frac{Z_1}{Z_4}\right) + \arg\left(\frac{Z_2}{Z_3}\right) = \arg\left(\frac{Z_1 \cdot Z_2}{Z_3 \cdot Z_4}\right) \]

4. **Calculating the Product**:
- Now, calculate the product \( Z_1 \cdot Z_2 \) and \( Z_3 \cdot Z_4 \):
\[ Z_1 \cdot Z_2 = (a + bi)(a - bi) = a^2 + b^2 \]
\[ Z_3 \cdot Z_4 = (c + di)(c - di) = c^2 + d^2 \]

5. **Final Argument Calculation**:
- Therefore, we have:
\[ \arg\left(\frac{Z_1 \cdot Z_2}{Z_3 \cdot Z_4}\right) = \arg\left(\frac{a^2 + b^2}{c^2 + d^2}\right) \]
- Since \( a^2 + b^2 \) and \( c^2 + d^2 \) are both real and positive, their argument is \( 0 \).

6. **Conclusion**:
- Thus, we conclude that:
\[ \arg\left(\frac{Z_1}{Z_4}\right) + \arg\left(\frac{Z_2}{Z_3}\right) = 0 \]

### Final Answer:
\[ \arg\left(\frac{Z_1}{Z_4}\right) + \arg\left(\frac{Z_2}{Z_3}\right) = 0 \]