Graph of ( z - (-1 - i) 2 ) in the Complex Plane

The equation ( z - (-1 - i) 2 ) describes a circle in the complex plane. Let's explore this concept step-by-step to understand the graphical representation better.

Introduction to Complex Numbers and the Problem

The complex plane is a geometric representation of complex numbers, where the real part corresponds to the x-axis and the imaginary part corresponds to the y-axis. The given equation, ( z - (-1 - i) 2 ), represents the set of all complex numbers whose distance from the point (-1 - i) is equal to 2. This is a fundamental idea in complex analysis, which is closely related to the concept of distance in the complex plane.

Breaking Down the Equation

Let's represent a complex number ( z ) as ( x yi ), where ( x ) and ( y ) are real numbers. Then, we can rewrite the given equation:

[ z - (-1 - i) 2 ]

[ x yi - (-1 - i) 2 ]

[ x 1 yi 1 2 ]

[ (x 1) (y 1)i 2 ]

Since the magnitude of a complex number ( a bi ) is ( sqrt{a^2 b^2} ), we rewrite:

[ sqrt{(x 1)^2 (y 1)^2} 2 ]

Squaring both sides, we get:

[ (x 1)^2 (y 1)^2 4 ]

Understanding the Equation

The equation ((x 1)^2 (y 1)^2 4) represents a circle in the ( xy )-plane, which is the complex plane. Let's break it down further:

Center: The center of the circle is at ((-1, -1)), which corresponds to the complex number (-1 - i).

Radius: The radius of the circle is 2.

Graphical Representation

The graph of ( z - (-1 - i) 2 ) is a circle centered at (-1 - i) with a radius of 2. In the complex plane, every point ( z ) on this circle satisfies the distance condition:

The distance from any point ( z ) on the circle to the point (-1 - i) is 2.

Summary

In conclusion, the equation ( z - (-1 - i) 2 ) in the complex plane describes a circle centered at (-1 - i) with a radius of 2. This is a direct application of the concept of distance in the complex plane and the geometric interpretation of complex numbers.

Final Answer: - Center: (-1 - i) - Radius: 2