If z is any complex number satisfying `abs(z-3-2i) le 2`, where `i=sqrt(-1)`, then the minimum value of `abs(2z-6+5i)`, is

Answer

To solve the problem step by step, we need to analyze the given condition and find the minimum value of the expression.

### Step-by-Step Solution:

1. **Understanding the Condition**:
We are given that \( |z - (3 + 2i)| \leq 2 \). This means that the complex number \( z \) lies within or on the boundary of a circle in the complex plane with center at \( (3, 2) \) and radius \( 2 \).

2. **Expressing \( z \)**:
Let \( z = x + yi \), where \( x \) and \( y \) are real numbers. Then the condition becomes:
\[ |(x + yi) - (3 + 2i)| \leq 2 \]
This simplifies to:
\[ |(x - 3) + (y - 2)i| \leq 2 \]
Which can be expressed as:
\[ \sqrt{(x - 3)^2 + (y - 2)^2} \leq 2 \]

3. **Finding the Region**:
The inequality describes a circle centered at \( (3, 2) \) with a radius of \( 2 \). Thus, \( x \) varies from \( 1 \) to \( 5 \) and \( y \) varies from \( 0 \) to \( 4 \).

4. **Expression to Minimize**:
We need to find the minimum value of \( |2z - 6 + 5i| \). Substituting \( z = x + yi \):
\[ |2(x + yi) - 6 + 5i| = |(2x - 6) + (2y + 5)i| \]
This can be expressed as:
\[ \sqrt{(2x - 6)^2 + (2y + 5)^2} \]

5. **Setting Up the Minimum Condition**:
To minimize \( \sqrt{(2x - 6)^2 + (2y + 5)^2} \), we can minimize the squared expression:
\[ (2x - 6)^2 + (2y + 5)^2 \]

6. **Finding Critical Points**:
We can find the minimum by analyzing the critical points:
- Set \( 2x - 6 = 0 \) which gives \( x = 3 \).
- Set \( 2y + 5 = 0 \) which gives \( y = -\frac{5}{2} \) (not possible since \( y \) must be between \( 0 \) and \( 4 \)).

7. **Evaluating at Valid Points**:
Since \( x = 3 \) is valid, we can evaluate at \( y = 0 \):
\[ |2(3) - 6 + 5(0)i| = |0 + 5i| = 5 \]

8. **Conclusion**:
The minimum value of \( |2z - 6 + 5i| \) is \( 5 \).

### Final Answer:
The minimum value of \( |2z - 6 + 5i| \) is \( 5 \).