Problem 8 - Advanced Engineering Mathematics

Question

In Problems 1-8, sketch the graph of the given equation. $$ |z+2+2 i|=2 $$

Step-by-Step Solution

Answer

The graph is a circle centered at \((-2, -2)\) with radius 2.

Step 1: Recognize Geometric Representation

In the complex plane, an equation of the form \(|z - z_0| = r\) represents a circle centered at point \(z_0\) with a radius \(r\). Here, \(z = x + yi\) and \(z_0 = -2 - 2i\), so the equation \(|z + 2 + 2i| = 2\) describes a circle centered at \((-2, -2)\) with a radius of 2.

Step 2: Convert Equation to Graphical Form

Rewrite \(|z + 2 + 2i| = 2\) using the distance formula. Given \(z = x + yi\), the equation becomes \(|(x + yi) + 2 + 2i| = 2\). Simplifying the expression, we get \(\sqrt{(x+2)^2 + (y+2)^2} = 2\). This is now the standard form of a circle's equation: \((x + 2)^2 + (y + 2)^2 = 4\).

Step 3: Determine the Center and Radius

From the equation \((x + 2)^2 + (y + 2)^2 = 4\), we identify the center as \((-2, -2)\) and the radius as \(2\). This tells us exactly where to position the circle on the complex plane.

Step 4: Sketch the Circle

Draw the complex plane with axes labeled as \(x\) (real part) and \(y\) (imaginary part). Place the center of the circle at \((-2, -2)\). Use the identified radius of \(2\) to draw a circle around this center. Verify the circle passes through points like \((0, -2)\) and \((-2, 0)\) to ensure accuracy.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane used to graphically represent complex numbers. In this plane, every complex number has a real part and an imaginary part. These parts correspond to the x-coordinate and y-coordinate on the plane, respectively.

The horizontal axis, known as the real axis, represents real numbers. The vertical axis, known as the imaginary axis, represents imaginary numbers. Complex numbers have the form \(z = x + yi\) where \(x\) is the real part and \(y\) is the imaginary part.

Visualizing complex numbers on this plane makes it easier to analyze certain properties, such as magnitude and direction. For example, the magnitude of a complex number \(z\) is its distance from the origin, calculated using \(|z| = \sqrt{x^2 + y^2}\). This geometric approach helps when working with complex equations, especially those representing shapes like circles.

Geometric Representation

In the complex plane, a geometric representation of an equation provides a visual understanding of its structure. Let's consider the equation \(|z - z_0| = r\). This expression describes a circle.

In this equation:

• \(z\) is a complex number variable.
• \(z_0\) is the center of the circle, a fixed complex number.
• \(r\) is the radius of the circle, a non-negative real number.

This structural form translates to the geometric representation of a circle centered at \(z_0\) with radius \(r\).

For example, \(|z + 2 + 2i| = 2\) becomes a circle centered at \((-2, -2)\) with a radius of \(2\). By rewriting this: first express \(z\) as \(x + yi\), then the formula \(|(x+2) + (yi+2i)| = 2\) confirms geometrically as a circle. This visualization aids in understanding how complex numbers form shapes on the plane.

Circle Equation

A circle's equation in the complex plane is derived from the general expression \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Given our problem, the transformation from \(|z + 2 + 2i| = 2\) to \((x + 2)^2 + (y + 2)^2 = 4\) illustrates this concept clearly.

• The formula shows the circle's center at \((-2, -2)\), simplifying how we locate the circle on the graph.
• The equation illuminates the radius as \(2\), which is essential for drawing and understanding the circle's size.

Mathematically, transforming a complex number given by \(z\) into this standard form utilizes the principle of distance, as \(|z-z_0|\) represents the radius. This standard circle equation is crucial for identifying specific features of circles formed by complex numbers, enhancing both graphical understanding and exact calculations.