`abs(z*omega)=1 ,arg(z)-arg(omega)=(3pi)/2`. Find the `arg[(1-2barzomega)/(1+3barzomega)]`

Options

A. `pi/4`
B. `-pi/4`
C. `(3pi)/4`
D. `-(3pi)/4`

Answer

To solve the problem, we need to find the argument of the complex expression given the conditions on \( z \) and \( \omega \). Let's break this down step by step.

### Step 1: Understand the Given Conditions
We are given:
1. \( |z \cdot \omega| = 1 \)
2. \( \arg(z) - \arg(\omega) = \frac{3\pi}{2} \)

From the first condition, we can deduce that:
- \( |z| \cdot |\omega| = 1 \)
- This implies \( |z| = \frac{1}{|\omega|} \).

### Step 2: Express \( z \) and \( \omega \)
Let:
- \( z = r e^{i\theta} \) where \( r = |z| \) and \( \theta = \arg(z) \).
- From the condition \( |z| = \frac{1}{|\omega|} \), we can express \( \omega \) as:
\[ \omega = \frac{1}{r} e^{i(\theta - \frac{3\pi}{2})} \]

### Step 3: Find \( \bar{z} \) and \( \bar{\omega} \)
The conjugates are:
- \( \bar{z} = r e^{-i\theta} \)
- \( \bar{\omega} = r e^{i(\theta - \frac{3\pi}{2})} \)

### Step 4: Calculate \( z \bar{\omega} \)
Now, we calculate \( z \bar{\omega} \):
\[ z \bar{\omega} = (r e^{i\theta}) \left( \frac{1}{r} e^{i(\theta - \frac{3\pi}{2})} \right) = e^{i\theta} e^{i(\theta - \frac{3\pi}{2})} = e^{i(2\theta - \frac{3\pi}{2})} \]

### Step 5: Substitute into the Expression
We need to find:
\[ \arg\left(\frac{1 - 2\bar{z}\omega}{1 + 3\bar{z}\omega}\right) \]
Calculating \( \bar{z} \omega \):
\[ \bar{z} \omega = (r e^{-i\theta}) \left( \frac{1}{r} e^{i(\theta - \frac{3\pi}{2})} \right) = e^{-i\theta} e^{i(\theta - \frac{3\pi}{2})} = e^{-i\frac{3\pi}{2}} = i \]

### Step 6: Substitute \( \bar{z} \omega \) into the Expression
Now substituting \( \bar{z} \omega = i \):
\[ \frac{1 - 2i}{1 + 3i} \]

### Step 7: Multiply by the Conjugate
To simplify, multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(1 - 2i)(1 - 3i)}{(1 + 3i)(1 - 3i)} = \frac{1 - 3i - 2i + 6}{1 + 9} = \frac{7 - 5i}{10} \]

### Step 8: Find the Argument
Now we find the argument of \( \frac{7 - 5i}{10} \):
- The argument is given by \( \tan^{-1}\left(\frac{-5}{7}\right) \).
- Since \( 7 > 0 \) and \( -5 < 0 \), this point lies in the fourth quadrant.

Thus, the argument is:
\[ \arg\left(\frac{7 - 5i}{10}\right) = \tan^{-1}\left(-\frac{5}{7}\right) \]

### Final Step: Adjust for Quadrant
The angle in the fourth quadrant can be expressed as:
\[ -\tan^{-1}\left(\frac{5}{7}\right) \]

### Conclusion
The final answer for the argument is:
\[ \arg\left(\frac{1 - 2\bar{z}\omega}{1 + 3\bar{z}\omega}\right) = -\tan^{-1}\left(\frac{5}{7}\right) \]