If z is any complex number satisfying `| z - 3- 2i | le 1 ` then the minimum value of `| 2 z - 6 + 5i|` is

Options

A. 5
B. 6
C. 7
D. 0

Answer

To solve the problem, we need to find the minimum value of \( |2z - 6 + 5i| \) given that \( |z - (3 + 2i)| \leq 1 \).

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
The condition \( |z - (3 + 2i)| \leq 1 \) describes a circle in the complex plane centered at the point \( (3, 2) \) with a radius of 1. This means that the complex number \( z \) can take any value within or on the boundary of this circle.

2. **Rewriting the Expression**:
We need to minimize \( |2z - 6 + 5i| \). We can rewrite this expression:
\[ |2z - 6 + 5i| = |2(z - 3 + \frac{5}{2}i)| \]
This simplifies to:
\[ 2|z - (3 - \frac{5}{2}i)| \]

3. **Finding the Center of the New Circle**:
The center of the circle for \( z \) is \( (3, 2) \) and we need to find the distance from this center to the point \( (3, -\frac{5}{2}) \).

4. **Calculating the Distance**:
The distance between the points \( (3, 2) \) and \( (3, -\frac{5}{2}) \) is calculated as follows:
\[ \text{Distance} = |2 - (-\frac{5}{2})| = |2 + \frac{5}{2}| = |2 + 2.5| = |4.5| = \frac{9}{2} \]

5. **Considering the Radius**:
Since the radius of the circle described by \( |z - (3 + 2i)| \leq 1 \) is 1, the minimum distance from the center \( (3, 2) \) to the point \( (3, -\frac{5}{2}) \) is:
\[ \text{Minimum distance} = \frac{9}{2} - 1 = \frac{7}{2} \]

6. **Final Calculation**:
Since we have \( 2|z - (3 - \frac{5}{2}i)| \), we multiply the minimum distance by 2:
\[ \text{Minimum value of } |2z - 6 + 5i| = 2 \times \frac{7}{2} = 7 \]

### Conclusion:
The minimum value of \( |2z - 6 + 5i| \) is \( 7 \).