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Step 1: Understand the Given Condition
We are given a set of complex numbers z such that |\,z - 2 - 2i\,|\le 1 .
Geometrically, this represents all points z in the complex plane whose distance from the point 2 + 2i is at most 1.
In other words, it is the disk (including its boundary) centered at (2,2) with radius 1.
Step 2: Identify the Expression to Maximize
We need to find the maximum value of:
|\,3iz + 6\,|.
Once we find the point z in the disk that maximizes this modulus, denoted by z = a + ib , we then need the sum a + b .
Step 3: Rewrite the Quantity in a More Convenient Form
Observe that:
3iz + 6 = 3i(z) + 6.
We can attempt to interpret 3iz geometrically. Multiplying by i in the complex plane rotates a point by 90^\circ counterclockwise and scales it if there's a coefficient.
Thus, 3iz can be viewed as taking the original point z , rotating by 90^\circ , and scaling by 3.
Step 4: Use a Geometric Interpretation
A standard approach is to interpret |\,3iz + 6\,| as the distance in the complex plane between the point 3iz and -6 .
But more directly, we can consider whether maximizing |\,3iz + 6\,| corresponds to selecting z on the boundary of the circle |\,z - (2 + 2i)\,|=1 .
Indeed, to maximize the distance from a fixed point (once the transformation is applied), we generally choose z on the boundary of the given circle.
Step 5: Find the Relevant Boundary Point
To locate the exact point, we can use the geometric fact that when looking for a maximum distance from a circle (after it undergoes a linear transformation in the complex plane), the maximum occurs on the circle boundary.
By evaluating or sketching (as hinted by the original solution), the point on the circle that maximizes the distance corresponds to z = 3 + 2i .
(One could perform an algebraic approach or use geometry to verify that this point indeed maximizes |\,3iz+6\,| .)
Step 6: Compute the Required Sum
Thus, the point where the maximum occurs is z = a + ib = 3 + 2i .
We need a + b , which is:
a + b = 3 + 2 = 5.
Final Answer
The value of a + b is 5.
