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Chapter 1: Problem 1

Sketch the graph of the given equation in the complex plane. $$ |z-4+3 i|=5 $$

Short Answer

Expert verified
The graph is a circle centered at \((4, -3)\) with radius 5.

Step by step solution

01

Understand the Equation

The equation \(|z - 4 + 3i| = 5\) is a geometric representation in the complex plane, where \(z\) is a complex number in the form \(z = x + yi\). Here, \(|z - 4 + 3i|\) represents the distance between the point \(z\) and the point \((4, -3)\) in the complex plane. The given equation indicates that this distance is constant and equals 5.
02

Interpret as a Circle

The equation \(|z - 4 + 3i| = 5\) implies that all the points \(z = x + yi\) are at a constant distance of 5 from the center point \((4, -3)\) in the complex plane. This is the definition of a circle with center \((4, -3)\) and radius 5.
03

Identify the Center and Radius

From the equation \(|z - 4 + 3i| = 5\), we identify the center of the circle to be \((4, -3)\). The radius of the circle is 5 units. These parameters are crucial for sketching the graph.
04

Sketch the Circle

To sketch the circle in the complex plane, plot the center at the point \((4, -3)\). Then, draw a circle around this center with a radius of 5. Ensure the circle extends 5 units in all directions from the center.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers
Complex numbers are numbers that have both a real part and an imaginary part. The general form is expressed as \( z = x + yi \), where:
  • \( x \) is the real part
  • \( y \) is the imaginary part
  • \( i \) is the imaginary unit, defined by the property \( i^2 = -1 \).
Understanding complex numbers is crucial for various fields such as engineering, physics, and mathematics. They allow for the representation and solving of problems that extend beyond the capability of real numbers alone. Whether it's simplifying complex equations or representing phenomena in electronics, complex numbers offer a broader numerical language.
Geometric Representation
The geometric representation of complex numbers involves visualizing them on the complex plane. The complex plane is similar to a coordinate plane, with:
  • The x-axis representing the real part of complex numbers
  • The y-axis representing the imaginary part
In this context, a complex number like \( z = x + yi \) is represented as a point \((x, y)\) on the complex plane. This visualization helps in understanding operations like addition or multiplication of complex numbers as geometric transformations. For instance, adding two complex numbers corresponds to adding their respective points on the plane.
Distance in the Complex Plane
The distance in the complex plane between two points \( z_1 = x_1 + y_1i \) and \( z_2 = x_2 + y_2i \) is given by the formula:\[|z_1 - z_2| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula is akin to the Euclidean distance formula in a two-dimensional space. It gives us a way to measure how far apart two complex numbers are within the plane. In the specific exercise, \(|z - 4 + 3i| = 5\), the expression \(|z - 4 + 3i|\) describes the distance from any point \( z \) to the center \((4, -3)\). The radius 5 indicates all points that consistently lie 5 units away from this center.
Circle in the Complex Plane
A circle in the complex plane is much like a circle in standard geometry. It consists of all points that are at a constant distance (radius) from a central point (center). Given the equation \(|z - (4 - 3i)| = 5\), we recognize immediately:
  • The center is the point \((4, -3)\)
  • The radius is 5
To sketch such a circle:1. **Plot the center** \((4, -3)\) on the complex plane.2. **Draw the circle** with the given radius of 5 units, ensuring it extends equally in all directions from the center.This setup allows for visualizing a locus of points that match the condition given by the equation, helping one grasp how complex numbers translate direct mathematical conditions into geometric shapes in the plane.

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Most popular questions from this chapter

Use de Moivre's formula (10) with \(n=2\) to find trigonometric identities for \(\cos 2 \theta\) and \(\sin 2 \theta\)

Sketch the set of points in the complex plane satisfying the given inequality. $$ 0 \leq \arg (z) \leq \pi / 6 $$

Consider the set \(S\) consisting of the complex plane with the circle \(|z|=5\) deleted. Give the boundary points of \(S .\) Is \(S\) connected?

Sketch the graph of the given equation in the complex plane. $$ |z+3 i|=2 $$

Solve the given pair of simultaneous equations. $$ |z|=2, \quad|z-2|=2 $$
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