Key Concepts and Formulas

• Modulus of a Complex Number: $|z|$ represents the distance of $z$ from the origin in the complex plane. $|z - z_0|$ represents the distance between $z$ and $z_0$.
• Geometric Interpretation of $|z - z_c| \le R$: This inequality represents a closed disk in the complex plane, centered at $z_c$ with radius $R$.
• Modulus Properties: $|zw| = |z||w|$ and $|z| = \sqrt{a^2 + b^2}$ where $z = a + bi$.

Step-by-Step Solution

Step 1: Interpret the given condition geometrically.

The given condition is $|z - 2 - 2i| \le 1$. We rewrite this as $|z - (2 + 2i)| \le 1$. This represents all complex numbers $z$ that lie within or on the boundary of a circle centered at $2 + 2i$ with radius 1. This is a disk in the complex plane.

Step 2: Transform the expression to be maximized.

We want to find the maximum value of $|3iz + 6|$. First, factor out $3i$: $|3iz + 6| = |3i(z + \frac{6}{3i})|$ Next, simplify $\frac{6}{3i}$: $\frac{6}{3i} = \frac{2}{i} = \frac{2}{i} \cdot \frac{-i}{-i} = \frac{-2i}{-i^2} = \frac{-2i}{1} = -2i$ Substitute this back into the expression: $|3i(z - 2i)| = |3i||z - 2i|$ Since $|3i| = \sqrt{0^2 + 3^2} = 3$, we have: $|3iz + 6| = 3|z - 2i|$ So, we want to maximize $|z - 2i|$.

Step 3: Determine the location of the center and point in the complex plane.

The region for $z$ is a disk centered at $z_c = 2 + 2i$ (corresponding to the point $(2, 2)$) with radius $R = 1$. We want to maximize the distance between $z$ and the point $P = 2i$ (corresponding to the point $(0, 2)$).

Step 4: Calculate the distance between the center of the disk and the point P.

The distance between the center of the disk $z_c = 2 + 2i$ and the point $P = 2i$ is: $|z_c - P| = |(2 + 2i) - 2i| = |2| = 2$ Since this distance (2) is greater than the radius of the disk (1), the point $P$ lies outside the disk.

Step 5: Find the point z that maximizes the distance.

The point $z$ that maximizes $|z - 2i|$ will lie on the boundary of the disk (the circle) and on the line connecting $2i$ to $2 + 2i$, extending outwards from $2 + 2i$. The vector from $2i$ to $2 + 2i$ is $(2 + 2i) - 2i = 2$. The unit vector in this direction is $\frac{2}{|2|} = 1$. Therefore, the complex number $z = a + ib$ where the maximum value is attained is: $z = z_c + R \cdot \frac{z_c - P}{|z_c - P|} = (2 + 2i) + 1 \cdot 1 = 3 + 2i$ Thus, $a = 3$ and $b = 2$.

Step 6: Calculate a + b.

The problem asks for the value of $a + b$: $a + b = 3 + 2 = 5$

Common Mistakes & Tips

• Incorrect Simplification: Be careful when simplifying fractions involving $i$. For example, $\frac{6}{3i} = -2i$.
• Geometric Visualization: Always visualize the complex plane to better understand the problem.
• Direction of Movement: Remember that when maximizing the distance from an external point, you move from the center of the circle away from the external point by the radius.

Summary

The problem involves maximizing the modulus of a complex expression subject to a constraint. We transformed the expression into a form that allowed us to use geometric interpretations. By identifying the region as a disk and finding the point on the disk farthest from a given external point, we were able to determine the complex number $z$ that maximizes the expression and calculate $a + b$. The final answer is 5.

Final Answer

The final answer is \boxed{5}.