Let z and `omega` be be complex numbers such that ` bar(z) + i omega = 0 and arg z omega = pi` then arg z =

Options

A. `pi//4`
B. `pi//2`
C. `3 pi//4`
D. ` 5 pi//4`

Answer

To solve the problem, we need to analyze the given conditions step by step.

### Step 1: Understand the given conditions
We have two conditions:
1. \( \bar{z} + i \omega = 0 \)
2. \( \arg(z) + \arg(\omega) = \pi \)

### Step 2: Express \( \omega \) in terms of \( z \)
From the first condition, we can express \( \omega \):
\[ \bar{z} + i \omega = 0 \implies i \omega = -\bar{z} \implies \omega = -\frac{\bar{z}}{i} = i \bar{z} \]
This is because dividing by \( i \) is equivalent to multiplying by \( -i \).

### Step 3: Substitute \( \omega \) into the second condition
Now, substituting \( \omega \) into the second condition:
\[ \arg(z) + \arg(i \bar{z}) = \pi \]

### Step 4: Simplify \( \arg(i \bar{z}) \)
Using properties of arguments:
\[ \arg(i \bar{z}) = \arg(i) + \arg(\bar{z}) = \frac{\pi}{2} + \arg(z) \]
because \( \arg(\bar{z}) = \arg(z) \).

### Step 5: Set up the equation
Now substituting this back into the equation:
\[ \arg(z) + \left(\frac{\pi}{2} + \arg(z)\right) = \pi \]
This simplifies to:
\[ 2\arg(z) + \frac{\pi}{2} = \pi \]

### Step 6: Solve for \( \arg(z) \)
Rearranging gives:
\[ 2\arg(z) = \pi - \frac{\pi}{2} = \frac{\pi}{2} \]
Thus,
\[ \arg(z) = \frac{\pi}{4} \]

### Final Answer
The value of \( \arg(z) \) is:
\[ \arg(z) = \frac{\pi}{4} \]

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