Suppose z and `omega` are two complex number such that `|z + iomega| = 2`. Which of the following is ture about `|z|` and `|omega|`?

Options

A. `|z|=|omega|= (1)/(2)`
B. `|z|=(1)/(2),|omega|,|omega| = (3)/(4)`
C. `|z| = |omega| = (3)/(4)`
D. `|z| = |omega| = 1`

Explanation

To solve the problem, we need to analyze the given condition involving the complex numbers \( z \) and \( \omega \).

### Step-by-Step Solution:

1. **Given Condition**: We have the condition \( |z + i\omega| = 2 \).

2. **Applying the Triangle Inequality**: According to the triangle inequality for complex numbers, we know that:
\[ |z + i\omega| \leq |z| + |i\omega| \]
Since \( |i\omega| = |\omega| \) (because the modulus of a complex number is independent of its argument), we can rewrite this as:
\[ |z + i\omega| \leq |z| + |\omega| \]

3. **Substituting the Given Condition**: From the given condition, we know that:
\[ 2 \leq |z| + |\omega| \]

4. **Rearranging the Inequality**: This implies that:
\[ |z| + |\omega| \geq 2 \]

5. **Analyzing the Options**: We need to determine which of the following statements about \( |z| \) and \( |\omega| \) is true based on the inequality we derived.

- The inequality \( |z| + |\omega| \geq 2 \) suggests that the sum of the moduli of \( z \) and \( \omega \) is at least 2. This means that neither \( |z| \) nor \( |\omega| \) can be less than 0, and at least one of them must be greater than or equal to 2.

6. **Conclusion**: The correct statement regarding \( |z| \) and \( |\omega| \) is that at least one of them must be greater than or equal to 2, or they can both be positive such that their sum is at least 2.