Key Concepts and Formulas

• The expression $|z_1 - z_2|$ represents the distance between two complex numbers $z_1$ and $z_2$ in the Argand plane.
• $|z|$ represents the distance of the complex number $z$ from the origin $(0,0)$ in the Argand plane.
• The inequality $|z - z_0| \le R$ represents a closed disk in the Argand plane with center $z_0$ and radius $R$.

Step-by-Step Solution

1. Geometric Interpretation of the Inequality

• What it means: We are given the inequality $|z - (3 - 2i)| \le 4$. This means that the distance between the complex number $z$ and the complex number $3 - 2i$ is less than or equal to 4.
• Why this step: Recognizing the geometric representation allows us to visualize the problem in the Argand plane, which simplifies finding the maximum and minimum values of $|z|$.
• The inequality represents a disk (circle and its interior) centered at $C = 3 - 2i$ (corresponding to the point $(3, -2)$ in the Argand plane) with radius $R = 4$.

2. Understanding the Objective Function

• What it means: We want to find the greatest and least values of $|z|$, where $|z|$ is the distance of the complex number $z$ from the origin $O(0, 0)$.
• Why this step: This clarifies that we are seeking the maximum and minimum distances from the origin to any point within the disk defined in Step 1.

3. Finding the Center's Distance from the Origin

• What it means: We need to find the distance between the center of the circle, $C(3, -2)$, and the origin, $O(0, 0)$.
• Why this step: This distance, along with the radius, will help us determine the maximum and minimum distances of $|z|$.
• We calculate the distance $OC$ as: $OC = \sqrt{(3 - 0)^2 + (-2 - 0)^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}$

4. Determining the Maximum and Minimum Values of |z|

• What it means: We need to find the points on the circle's boundary that are farthest from and closest to the origin. Since the origin may be inside or outside the circle, we need to consider both scenarios.

• Why this step: The maximum and minimum distances from the origin to the disk will occur along the line connecting the origin and the center of the circle. Since $OC = \sqrt{13} \approx 3.6 < 4 = R$, the origin lies inside the circle.

• Maximum Value of $|z|$: The maximum distance is the distance from the origin to the center plus the radius: $|z|_{\text{max}} = OC + R = \sqrt{13} + 4$

• Minimum Value of $|z|$: Since the origin is inside the circle, the minimum distance from the origin to the boundary of the circle is the radius minus the distance from the origin to the center: $|z|_{\text{min}} = R - OC = 4 - \sqrt{13}$

5. Calculating the Difference

• What it means: We need to find the difference between the maximum and minimum values of $|z|$.
• Why this step: This gives us the final answer to the problem.
• The difference is: $|z|_{\text{max}} - |z|_{\text{min}} = (4 + \sqrt{13}) - (4 - \sqrt{13}) = 4 + \sqrt{13} - 4 + \sqrt{13} = 2\sqrt{13}$

Common Mistakes & Tips

• Visualize: Always draw a diagram to represent the complex numbers and the region in the Argand plane.
• Origin Inside/Outside: Carefully determine whether the origin lies inside or outside the circle. If it's inside, the minimum distance is $R - OC$; if it's outside, the minimum distance is $OC - R$.
• Triangle Inequality: Remember that the triangle inequality can be helpful in determining the bounds on $|z|$.

Summary

The problem involves finding the difference between the maximum and minimum values of $|z|$ given the constraint $|z - (3 - 2i)| \le 4$. By interpreting the constraint geometrically as a disk in the Argand plane, we found the distance between the origin and the center of the disk, determined that the origin lies inside the disk, and calculated the maximum and minimum distances from the origin to the boundary of the disk. The difference between these distances is $2\sqrt{13}$.

Final Answer The final answer is \boxed{2\sqrt {13}}, which corresponds to option (A).