Minimum Value of Complex Number Expression in Region

To find the minimum value of the expression $|2z - 6 + 5i|$ given that $|z - 3 - 2i| \leq 2$, we can use geometric interpretation in the Argand plane.

Step 1: Understand the given region
The inequality
$|z - (3 + 2i)| \leq 2$
represents a closed circular disk in the complex plane centered at $z_0 = 3 + 2i$ with a radius of $r = 2$.

Step 2: Rewrite the expression to find the minimum value
We want to find the minimum value of:
$E = |2z - 6 + 5i|$

Factoring out $2$ from inside the modulus gives:
$E = 2 \left| z - \left( 3 - \frac{5}{2}i \right) \right|$

Let $z_1 = 3 - \frac{5}{2}i$. Thus, the expression $E$ represents twice the distance from the variable complex number $z$ to the fixed point $z_1$:
$E = 2 |z - z_1|$

Step 3: Calculate the distance between the center $z_0$ and the point $z_1$
The distance between the center of the disk $z_0 = 3 + 2i$ and the point $z_1 = 3 - \frac{5}{2}i$ is:
$|z_0 - z_1| = \left| (3 + 2i) - \left( 3 - \frac{5}{2}i \right) \right| = \left| \left( 2 + \frac{5}{2} \right) i \right| = \left| \frac{9}{2}i \right| = \frac{9}{2}$

Step 4: Determine the minimum distance
Since the distance from the center to $z_1$ is $\frac{9}{2} = 4.5$, which is strictly greater than the radius $r = 2$, the point $z_1$ lies outside the circular disk.

The minimum distance from any point $z$ inside or on the boundary of the disk to the point $z_1$ occurs along the line segment connecting $z_0$ and $z_1$, and is given by:
$|z - z_1|_{\text{min}} = |z_0 - z_1| - r = \frac{9}{2} - 2 = \frac{5}{2}$

Step 5: Calculate the minimum value of the expression
Substituting this minimum distance back into our expression for $E$:
$E_{\text{min}} = 2 \times \frac{5}{2} = 5$

Thus, the minimum value of $|2z - 6 + 5i|$ is $5$.