If ` z_(1) , z_(2)` are conjugate complex numbers, and `z_(3) , z_(4)` are also conjugate, then arg ` (z_(3))/(z_(2))`

Options

A. `arg "" (z_(1))/(z_(4))`
B. `arg"" (z_(4))/( z_(1))`
C. `arg"" (z_(2))/(z_(4))`
D. `arg "" (z_(1))/( z_(3))`

Answer

To solve the problem, we need to find the argument of the quotient of two complex numbers, specifically \( \arg\left(\frac{z_3}{z_2}\right) \), given that \( z_1 \) and \( z_2 \) are conjugate complex numbers, and \( z_3 \) and \( z_4 \) are also conjugate complex numbers.

### Step-by-Step Solution:

1. **Understanding Conjugate Complex Numbers**:
- If \( z_1 = a + bi \), then its conjugate \( z_2 = a - bi \).
- Similarly, if \( z_3 = c + di \), then its conjugate \( z_4 = c - di \).

2. **Expressing the Quotient**:
- We need to find \( \frac{z_3}{z_2} \).
- Using the definitions, we can write:
\[ \frac{z_3}{z_2} = \frac{c + di}{a - bi} \]

3. **Multiplying by the Conjugate**:
- To simplify this expression, we multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{z_3}{z_2} = \frac{(c + di)(a + bi)}{(a - bi)(a + bi)} = \frac{(ca + bdi + abi - db)}{a^2 + b^2} \]
- This simplifies to:
\[ \frac{(ca + bd) + (ad - bc)i}{a^2 + b^2} \]

4. **Finding the Argument**:
- The argument of a complex number \( x + yi \) is given by \( \tan^{-1}\left(\frac{y}{x}\right) \).
- Thus, we have:
\[ \arg\left(\frac{z_3}{z_2}\right) = \tan^{-1}\left(\frac{ad - bc}{ca + bd}\right) \]

5. **Using Properties of Conjugates**:
- Since \( z_2 \) is the conjugate of \( z_1 \), we also know:
\[ \arg(z_2) = -\arg(z_1) \]
- Similarly, since \( z_4 \) is the conjugate of \( z_3 \):
\[ \arg(z_4) = -\arg(z_3) \]

6. **Final Expression**:
- Therefore, we can conclude:
\[ \arg\left(\frac{z_3}{z_2}\right) = \arg(z_3) - \arg(z_2) = \arg(z_3) + \arg(z_1) \]
- Since \( z_3 \) and \( z_4 \) are conjugates, we have:
\[ \arg(z_3) = -\arg(z_4) \]
- Thus, the final result is:
\[ \arg\left(\frac{z_3}{z_2}\right) = \arg(z_3) + \arg(z_1) \]

### Conclusion:
The argument \( \arg\left(\frac{z_3}{z_2}\right) \) can be expressed in terms of the arguments of the conjugate pairs.