Problem 8 - Advanced Engineering Mathematics

Question

Sketch the graph of the given equation. $$ |z+2+2 i|=2 $$

Step-by-Step Solution

Answer

The equation represents a circle centered at \(-2 - 2i\) with radius 2.

Step 1: Interpret the Equation

The equation \(|z+2+2i| = 2\) represents a circle in the complex plane centered at \(-2 - 2i\), with a radius of 2 units. This is because the modulus \(|z - a| = r\) describes a circle centered at the point \(a\) with radius \(r\).

Step 2: Determine the Center

The center of the circle can be found by equating \(z + 2 + 2i\) with \((x + yi)\) where \(x, y\) are real numbers, giving us the center \(-2 - 2i\) on the complex plane.

Step 3: Determine the Radius

The radius of the circle is given directly by the right-hand side of the equality, which is 2.

Step 4: Sketch the Graph

On a 2D plane with the real axis as the x-axis and the imaginary axis as the y-axis, plot the center of the circle at \(-2, -2\). Then, sketch a circle with a radius of 2 around this center.

Key Concepts

Circle Equation

In mathematics, a circle is often defined by an equation. On the complex plane, this equation usually takes the form

• \(|z - a| = r\)

Here, \(z\) is a complex number, \(a\) indicates the center of the circle, and \(r\) represents the radius. To break it down, this equation signifies all points \(z\) that are exactly \(r\) units away from \(a\).
Understanding the relation between a complex number equation and circles is crucial. In the equation \(|z+2+2i|=2\), notice how it fits the template \(|z - a| = r\). You can see that it describes a circle with its center at

• \(-2 - 2i\)

and a radius given on the right side, which is 2.
The equation of a circle on the complex plane is very useful. It helps identify important characteristics like the center and radius, and allows us to visualize its position and size on the plane.

Modulus of a Complex Number

The modulus of a complex number provides a measure of its distance from the origin on the complex plane. For a complex number \(z = x + yi\) where \(x\) is the real part and \(y\) is the imaginary part, the modulus is calculated as:

• \(|z| = \sqrt{x^2 + y^2}\)

The modulus is always a non-negative value and represents the "length" or "magnitude" of the vector from the origin to the point \((x, y)\) on the plane.
When dealing with the circle equation in the complex plane, the modulus plays a central role. It helps define the radius by making sure the distance is constant from the center to any point on the circumference. In our example, the modulus \(|z + 2 + 2i| = 2\) shows that every point \(z\) on the circle satisfies this constant radius condition of 2 units from the center \(-2 - 2i\).
Grasping the modulus concept is key when working with complex numbers and equations, specifically in scenarios involving geometrical interpretations on the complex plane.

Graph Sketching

Graph sketching in the complex plane involves visually representing equations, helping us understand their geometric properties. To sketch the graph of \(|z + 2 + 2i| = 2\), you first need to identify the main elements: center and radius.
In this case, we established that our circle is centered at \(-2, -2\) and has a radius of 2. To represent this correctly:

• Begin with plotting the center
  • \(-2, -2\)
  as a point on the complex plane
• From the center, draw a circle with a set radius of 2.
• Use the real axis as the x-axis and the imaginary axis as the y-axis for accurate orientation.

This visual sketch provides clarity by confirming the circle's location and size within the scope of the complex plane. Regular practice with graph sketching will enhance your understanding of how complex equations can look geographically, and how their parameters influence their shapes and placements on the plane.