A point in the region |z-3| ≤2 such that arg z0 is a maximum. (a) (5)/(3)+(2 √(5))/(3) i (b) (1)

step 1 In this question, we need to find a point in the reason. Modulus z minus 3 less than equals to 2

A point in the region |z-3| ≤2 such that arg z0 is a...

A point in the region $|z-3| \leq 2$ such that arg $z_{0}$ is a maximum.
(a) $\frac{5}{3}+\frac{2 \sqrt{5}}{3} \mathrm{i}$
(b) $\frac{1}{3}+\frac{2}{3} \mathrm{i}$
(c) $5+\frac{2}{3}$ i
(d) $\frac{\sqrt{5}}{3}+\frac{2}{3} \mathrm{i}$

Key Concepts

Geometric Optimization in Complex Analysis

Maximizing or minimizing functions related to complex numbers within a given region often involves geometric reasoning. Such problems require identifying extreme points on boundaries of regions like disks or circles where quantities like the argument or modulus achieve their maximum or minimum values.

Complex Number Geometry

Complex numbers can be represented as points on the complex plane, where the horizontal axis corresponds to the real part and the vertical axis to the imaginary part. This provides a geometric way to interpret operations on complex numbers, such as addition and multiplication, and to visualize properties like magnitude and argument.

Regions in the Complex Plane

Inequalities like |z - a| ? r define regions in the complex plane, usually circular disks. These regions represent all points whose distance from a specific point (the center) is within a certain radius, which is crucial for analyzing and solving optimization problems involving complex numbers.

Argument of a Complex Number

The argument of a complex number is the angle between the positive real axis and the line connecting the origin to the point representing the complex number. It is a fundamental concept in understanding the orientation of complex numbers and plays a key role in many problems involving angular measurements or directional constraints.

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