Problem 19 - Contemporary Precalculus

Question

In Exercises \(17-24,\) sketch the graph of the equation in the complex plane (z denotes a complex number of the form a \(+b i\) ). \(|z-1|=10[\text {Hint: } 1 \text { corresponds to }(1,0)\) in the complex plane. What does the equation say about the distance from \(z\) to \(1 ?]\)

Step-by-Step Solution

Answer

Answer: The graph of the given complex equation \(|z-1|=10\) is a circle in the complex plane, centered at (1,0) with a radius of 10 units.

Step 1: Understand the Meaning of the Equation

The equation \(|z-1|=10\) represents all the complex numbers z whose distance from the complex number 1 (corresponding to the point (1,0)) is equal to 10. In other words, it represents a circle with a center at (1,0) and a radius of 10.

Step 2: Find the Equation of the Circle in the Complex Plane

To find the equation of the circle in the complex plane, we can rewrite the complex number z as \(a+bi\). Then, the equation \(|z-1|=10\) becomes \(|a+bi-1|=10\). Simplify the equation to \(|a-1+bi|=10\). Now, we can use the distance formula for a complex number, which is given by \(|a-1+bi|=sqrt((a-1)^2+b^2)\). Set the distance equal to 10: \(10=\sqrt{(a-1)^2+b^2}\) Square both sides of the equation: \(100=(a-1)^2+b^2\) This is the equation of the circle in the complex plane.

Step 3: Sketch the Graph of the Equation in the Complex Plane

We now have the equation of the circle, \(100=(a-1)^2+b^2\). To sketch the graph, follow these steps: 1. Locate the center of the circle, which is given as (1,0). 2. Draw a circle with the radius of 10 units from the center (1,0). 3. Label the circle with its equation, \(100=(a-1)^2+b^2\), where z=a+bi. The resulting graph is a circle centered at (1,0) with a radius of 10 in the complex plane. All the points on this circle are complex numbers that satisfy the given equation, \(|z-1|=10\).

Key Concepts

Complex Plane

The complex plane is a uniquely useful way of representing complex numbers. Each complex number \( z = a + bi \) can be plotted as a point where \( a \) is the real part on the horizontal axis and \( b \) is the imaginary part on the vertical axis. This forms a coordinate system similar to the Cartesian plane, but it's specific to complex numbers. When you see a point like \((1, 0)\), it indicates a complex number with a real part of 1 and an imaginary part of 0. Complex numbers are typically visualized as specific points in this plane, aiding in their understanding and manipulation.

• The horizontal axis represents the real part of complex numbers.
• The vertical axis represents the imaginary part of complex numbers.

Understanding how to plot a complex number on the complex plane is the first step in analyzing equations that involve these numbers. This graphical representation provides a powerful method for visualizing operations such as addition, subtraction, and finding distances.

Distance Formula

In the context of the complex plane, the distance formula is a tool to find the separation between two points, or complex numbers. If you have a complex number \( z = a + bi \) and another complex number \( w = c + di \), the distance between these two points in the complex plane can be calculated using:\[|z - w| = \sqrt{(a - c)^2 + (b - d)^2}\]For example, in the equation \(|z-1| = 10\), we are focused on complex numbers \( z \) whose distance from the complex number \( 1 \) (represented on the complex plane as \((1, 0)\)) is exactly 10 units. This forms a circular region around \( (1, 0) \).

• The formula gives you the straight-line distance, often referred to as the Euclidean distance.
• This concept can be applied to various problems, providing a geometric interpretation of complex operations.

Graphing

Graphing in the context of complex numbers is essentially plotting on the complex plane. This process allows you to visualize equations graphically. When you plot the complex number \( z = a + bi \), you're placing a point at the coordinates \((a, b)\).To sketch an equation like \(|z-1| = 10\), you will:

• Identify the center of the circle \((1, 0)\).
• Draw a circle with a radius equal to the specified distance, which is 10 units in this case.

This visual representation can help in understanding the underlying geometric nature of equations involving complex numbers, allowing you to see, at a glance, the set of all complex numbers that satisfy a given condition.

Circle Equation

A circle equation in the complex plane like \(|z-1| = 10\) has its unique interpretation. This equation represents all complex numbers \( z \) that lie on a circle centered at the point corresponding to \( 1 \), or \((1, 0)\) on the complex plane, with a radius of 10.The equation can be transformed into a more familiar form by expressing \( z \) as \( a + bi \), leading to:\[ |a - 1 + bi| = 10 \\sqrt{(a-1)^2 + b^2} = 10 \100 = (a-1)^2 + b^2 \]This gives us the standard form of a circle equation in the coordinate plane, where \((a-1)^2 + b^2\) indicates a circle centered at \((1, 0)\) with radius 10.

• The \(|z - 1| = 10\) format is a concise way to express this in the complex domain.
• It provides geometry-intuitive insights for analyzing distances and boundaries.