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Magazine Chapter 3: Problem 49
Given that the argument of \((z-1)(z+1)\) is \(\frac{1}{4} \pi\), show that the locus of \(a\) in the Argand diagram is part of a circle of centre \((0,1)\) and radius \(\sqrt{2}\).
Short Answer
Expert verified
The locus of \(z\) is part of a circle centered at \((0,1)\) with radius \(\sqrt{2}\).
Step by step solution
01
Understanding the Problem
We start with the expression \((z-1)(z+1)\) and know that its argument is \(\frac{1}{4} \pi\). We aim to show that the locus of points \(z\) describes a circle in the Argand diagram with center \((0,1)\) and radius \(\sqrt{2}\).
02
Set z as a Complex Number
Let \(z = a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. This represents a generic complex number \(z\).
03
Express z-1 and z+1
We replace \(z\) in the expressions \(z-1\) and \(z+1\): \(z-1 = (a-1) + bi\) and \(z+1 = (a+1) + bi\).
04
Multiply the Complex Numbers
Compute \((z-1)(z+1)\): \[(z-1)(z+1) = ((a-1) + bi)((a+1) + bi) = (a^2-1) + 2abi + b^2\].
05
Determine the Argument
The argument of a complex number \(x + yi\) is given by \(\tan^{-1}\left(\frac{y}{x}\right)\). Thus, the argument of \((z-1)(z+1)\) is \[\tan^{-1}\left(\frac{2ab}{a^2+b^2-1}\right) = \frac{1}{4} \pi\].
06
Use the Tangent Identity
Since the argument equals \(\frac{1}{4} \pi\), we know that \(\tan\left(\frac{1}{4} \pi\right) = 1\). Set \[\frac{2ab}{a^2 + b^2 - 1} = 1\].
07
Simplify the Equation
Solve the equation \[2ab = a^2 + b^2 - 1\]which rearranges to \[a^2 - 2ab + b^2 = 1\]. This represents the equation of a circle.
08
Identify the Circle Equation
Recognize the derived equation \(a^2 - 2ab + b^2 = 1\) is equivalent to \[(a-b)^2 + (b-1)^2 = 2\]. This is the standard form of a circle centered at \((0,1)\) with radius \(\sqrt{2}\).
09
Conclusion
The locus of \(z\) is a circle with center at \((0,1)\) and radius \(\sqrt{2}\) in the Argand diagram.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Argand Diagram
The Argand Diagram is a visual tool used to represent complex numbers in a two-dimensional plane. On this diagram, the horizontal axis (the real axis) represents the real part of a complex number, while the vertical axis (the imaginary axis) represents the imaginary part.
- The complex number \(z = a + bi\) is plotted on the diagram as the point \((a, b)\).
- This makes it easier to perform arithmetic operations on complex numbers visually.
- In this plane, a circle or any geometric shape can represent the locus of a complex number satisfying a particular condition.
Locus
In mathematics, the term "locus" refers to a set of points that satisfy a particular condition. This condition can often be an equation or inequality involving the coordinates of the points.
- In the context of complex numbers, the locus represents all possible values of a complex number \(z\) that meet specific criteria.
- The exercise demonstrates how to find the locus of a complex number on an Argand Diagram by setting conditions on the argument of a derived expression.
- The result of these conditions is typically a recognizable shape, such as a line or circle, on the diagram.
Circle Equation
The equation of a circle is a fundamental concept in geometry, typically expressed in the form \[(x-h)^2 + (y-k)^2 = r^2\]where \((h,k)\) is the center, and \(r\) is the circle's radius.
- For complex numbers, the set of points forming a circle can be identified using similar equations.
- In the exercise, the equation \((a-b)^2 + (b-1)^2 = 2\) describes a circle centered at \((0,1)\) with radius \(\sqrt{2}\).
- This expression comes from setting the conditions that a product of complex number differences has a specific argument.
Imaginary Unit
The imaginary unit, denoted as \(i\), is a fundamental mathematical entity used to define complex numbers. It is defined by the property \(i^2 = -1\).
- Complex numbers take the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) represents an imaginary component.
- The inclusion of the imaginary unit allows for the representation of numbers that do not exist on the real number line.
- In our exercise, \(i\) is a crucial part of expressing and manipulating the complex numbers \(z-1\) and \(z+1\).
