If `|z+2-i|=5` and maxium value of `|3z +9-7i|` is M, then the value of M is ______.

Explanation

To solve the problem, we need to find the maximum value of \( |3z + 9 - 7i| \) given the condition \( |z + 2 - i| = 5 \).

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
We have the equation \( |z + 2 - i| = 5 \). This represents a circle in the complex plane centered at \( -2 + i \) with a radius of 5.

2. **Expressing the Complex Number**:
We can express \( z \) in terms of its real and imaginary parts:
\[ z = x + yi \]
where \( x \) and \( y \) are real numbers.

3. **Rewriting the Condition**:
The condition \( |z + 2 - i| = 5 \) can be rewritten as:
\[ |(x + 2) + (y - 1)i| = 5 \]
This implies:
\[ \sqrt{(x + 2)^2 + (y - 1)^2} = 5 \]
Squaring both sides gives:
\[ (x + 2)^2 + (y - 1)^2 = 25 \]

4. **Finding the Maximum Value**:
We need to find the maximum value of \( |3z + 9 - 7i| \):
\[ |3z + 9 - 7i| = |3(x + yi) + 9 - 7i| = |(3x + 9) + (3y - 7)i| \]
This can be expressed as:
\[ |(3x + 9) + (3y - 7)i| = \sqrt{(3x + 9)^2 + (3y - 7)^2} \]

5. **Using the Triangle Inequality**:
We can apply the triangle inequality:
\[ |3z + 9 - 7i| \leq |3(z + 2 - i)| + |9 - 7i| \]
Here, we know \( |z + 2 - i| = 5 \), so:
\[ |3(z + 2 - i)| = 3 \cdot 5 = 15 \]
Now, we calculate \( |9 - 7i| \):
\[ |9 - 7i| = \sqrt{9^2 + (-7)^2} = \sqrt{81 + 49} = \sqrt{130} \]

6. **Combining the Results**:
Therefore, we have:
\[ |3z + 9 - 7i| \leq 15 + \sqrt{130} \]

7. **Finding the Maximum Value**:
We need to find the maximum value of \( |3z + 9 - 7i| \). The maximum occurs when both components are maximized. Thus:
\[ M = 15 + \sqrt{130} \]

8. **Final Calculation**:
Since we need to find the numerical value of \( M \):
\[ \sqrt{130} \approx 11.4 \quad \text{(approximately)} \]
Therefore:
\[ M \approx 15 + 11.4 = 26.4 \]

### Conclusion:
The maximum value \( M \) is approximately \( 26.4 \).