In Q.no. 88, if z be any point in `A frown B frown C` and `omega` be any point satisfying `|omega-2-i| lt 3`. Then, `|z|-|omega|+3` lies between

Options

A. `-6` and 3
B. `-3` and 6
C. `-6` and 6
D. `-3` and 9

Explanation

To solve the problem, we need to analyze the given conditions and derive the required expression step by step.

### Step-by-Step Solution:

1. **Understanding the Given Conditions**:
We are given that \( \omega \) satisfies the condition \( |\omega - 2 - i| < 3 \). This describes a circle in the complex plane centered at the point \( (2, 1) \) with a radius of \( 3 \).

2. **Identifying the Points**:
Let \( z \) be any point in the triangle formed by points \( A \), \( B \), and \( C \). We denote the distance from point \( z \) to point \( \omega \) as \( |z - \omega| \).

3. **Using the Triangle Inequality**:
According to the triangle inequality, we have:
\[ ||z| - |\omega|| \leq |z - \omega| \leq |z| + |\omega| \]
This means that the difference in the magnitudes of \( z \) and \( \omega \) is bounded by the distance between the two points.

4. **Finding the Bounds for \( |z - \omega| \)**:
Since \( \omega \) lies inside the circle defined by \( |\omega - 2 - i| < 3 \), we can analyze the maximum and minimum values of \( |\omega| \):
- The center of the circle is at \( (2, 1) \), so the distance from the origin to the center is \( \sqrt{2^2 + 1^2} = \sqrt{5} \).
- The maximum distance from the origin to any point \( \omega \) in the circle is \( \sqrt{5} + 3 \).
- The minimum distance from the origin to any point \( \omega \) in the circle is \( \sqrt{5} - 3 \) (if \( \sqrt{5} > 3 \)).

5. **Calculating the Values**:
- Maximum value of \( |\omega| \):
\[ |\omega| < \sqrt{5} + 3 \]
- Minimum value of \( |\omega| \):
\[ |\omega| > \sqrt{5} - 3 \]

6. **Substituting into the Expression**:
We need to find \( |z| - |\omega| + 3 \). Using the bounds we found:
\[ |z| - (\sqrt{5} + 3) + 3 < |z| - |\omega| + 3 < |z| - (\sqrt{5} - 3) + 3 \]
Simplifying the inequalities gives:
\[ |z| - \sqrt{5} < |z| - |\omega| + 3 < |z| + 3 - \sqrt{5} \]

7. **Final Result**:
Therefore, the expression \( |z| - |\omega| + 3 \) lies between:
\[ |z| - \sqrt{5} \quad \text{and} \quad |z| + 3 - \sqrt{5} \]

### Conclusion:
The final answer is that \( |z| - |\omega| + 3 \) lies between \( |z| - \sqrt{5} \) and \( |z| + 3 - \sqrt{5} \).