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Magazine Chapter 17: Problem 7
Sketch the graph of the given equation. $$ |z-4+3 i|=5 $$
Short Answer
Expert verified
The graph is a circle centered at (4, -3) with radius 5.
Step by step solution
01
Understanding the Equation
The equation \(|z - 4 + 3i| = 5\) represents the set of all complex numbers \(z\) that are at a distance of 5 from the point \((4, -3)\) in the complex plane.
02
Identifying the Center
Recognize that the complex number \(z\) can be expressed as \(z = x + yi\). The expression \(|z - 4 + 3i|\) translates the problem to finding the distance from \(z = (x, y)\) to the point \((4, -3)\).
03
Circle Equation Formulation
Given the distance is 5, the equation \(|z - 4 + 3i| = 5\) can be rewritten as \(\sqrt{(x - 4)^2 + (y + 3)^2} = 5\), which is the equation of a circle.
04
Squaring Both Sides
Square both sides to eliminate the square root, resulting in \((x - 4)^2 + (y + 3)^2 = 25\).
05
Graphing the Circle
The equation \((x - 4)^2 + (y + 3)^2 = 25\) describes a circle with center at \((4, -3)\) and radius of 5 units. Graph this circle on the complex plane.
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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 two parts: a real part and an imaginary part. They are generally written in the form \(z = a + bi\), where \(a\) represents the real part and \(b\) represents the imaginary part multiplied by \(i\), which is the square root of -1.
Understanding complex numbers is crucial as they provide a way to represent and solve equations that do not have real solutions, especially in electronics and other fields of engineering.
Understanding complex numbers is crucial as they provide a way to represent and solve equations that do not have real solutions, especially in electronics and other fields of engineering.
- For example, the complex number \(3 + 4i\) has a real part \(3\) and an imaginary part \(4\).
Distance in Complex Plane
In the complex plane, a complex number \(z = x + yi\) is represented as a point \((x, y)\).
The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in this plane is given by \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
This distance formula is directly related to the modulus of a complex number.
The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in this plane is given by \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
This distance formula is directly related to the modulus of a complex number.
- The modulus of a complex number \(z = x + yi\) is \(|z| = \sqrt{x^2 + y^2}\).
Circle Equation
The equation of a circle in a coordinate plane can be identified by its standard form: \[(x - h)^2 + (y - k)^2 = r^2\]where \((h, k)\) is the center of the circle and \(r\) is the radius.
In complex numbers, a similar concept applies, where the expression \(|z - (h + ki)| = r\) forms a circle centered at \((h, -k)\) in the complex plane.
In complex numbers, a similar concept applies, where the expression \(|z - (h + ki)| = r\) forms a circle centered at \((h, -k)\) in the complex plane.
- For our exercise, it represents a circle with center at \((4, -3)\) and radius \(5\).
Graphing in Complex Plane
To graph a complex number on the complex plane, you treat the real part as the x-coordinate and the imaginary part as the y-coordinate. Visualizing complex numbers helps in understanding not only their mathematical operations but also their geometric significance.
When graphing a circle such as described by the exercise equation \[(x - 4)^2 + (y + 3)^2 = 25\], you would plot a circle centered at the point \((4, -3)\) with radius \(5\) units.
When graphing a circle such as described by the exercise equation \[(x - 4)^2 + (y + 3)^2 = 25\], you would plot a circle centered at the point \((4, -3)\) with radius \(5\) units.
- Start by plotting the center of the circle at \((4, -3)\).
- Then, mark a radius of 5 units around this point in all directions.
