Let `z and omega` be two non zero complex numbers such that `|z|=|omega|` and `argz+argomega=pi`, then z equals (A) `omega` (B) `-omega` (C) `baromega` (D) `-baromega`

Options

A. `omega`
B. `-omega`
C. `-bar(omega)`
D. `bar(omega)`

Answer

To solve the problem, we start with the given conditions about the complex numbers \( z \) and \( \omega \).

### Step 1: Understand the Given Conditions
We know that:
1. \( |z| = |\omega| \) (the magnitudes of \( z \) and \( \omega \) are equal).
2. \( \arg z + \arg \omega = \pi \) (the sum of the arguments of \( z \) and \( \omega \) is \( \pi \)).

### Step 2: Express the Complex Numbers
Let’s express \( z \) and \( \omega \) in their polar forms:
- \( z = r e^{i \theta} \)
- \( \omega = r e^{i \phi} \)

where \( r = |z| = |\omega| \) and \( \theta = \arg z \), \( \phi = \arg \omega \).

### Step 3: Use the Argument Condition
From the condition \( \arg z + \arg \omega = \pi \):
\[ \theta + \phi = \pi \]
This implies:
\[ \phi = \pi - \theta \]

### Step 4: Substitute the Argument into the Expression for \( \omega \)
Substituting \( \phi \) into the expression for \( \omega \):
\[ \omega = r e^{i(\pi - \theta)} = r (\cos(\pi - \theta) + i \sin(\pi - \theta)) = r (-\cos \theta + i \sin \theta) = -r e^{i \theta} = -z \]

### Step 5: Conclusion
Thus, we find that:
\[ z = -\omega \]

### Final Answer
The correct option is (B) \( -\omega \).

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