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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 \(z\) 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 a circle centered at (0, 1) with radius \(\sqrt{2}\).
Step by step solution
01
Introduce Complex Number Representation
Let the complex number \(z\) be represented as \(z = x + yi\), where \(x\) is the real part and \(y\) is the imaginary part. The given expression for the argument is \(\text{arg}\left(\frac{z-1}{z+1}\right) = \frac{1}{4}\pi\).
02
Rewrite Expression in Terms of x and y
Express \(z-1\) and \(z+1\) in terms of \(x\) and \(y\):\[(z-1)=(x-1)+yi\]\[(z+1)=(x+1)+yi\]Therefore, \(\frac{z-1}{z+1}\) becomes \[\frac{(x-1)+yi}{(x+1)+yi}\].
03
Find Argument of the Expression
The argument of a complex number \(\frac{a+bi}{c+di}\) is given by the formula: \[\text{arg}\left(\frac{a+bi}{c+di}\right) = \text{arg}(a+bi) - \text{arg}(c+di)\].Hence, \[\text{arg}\left((x-1)+yi\right) = \text{tan}^{-1}\left(\frac{y}{x-1}\right)\]and\[\text{arg}\left((x+1)+yi\right) = \text{tan}^{-1}\left(\frac{y}{x+1}\right)\].
04
Set Up the Equation
Using the argument property, we have \[\text{tan}^{-1}\left(\frac{y}{x-1}\right) - \text{tan}^{-1}\left(\frac{y}{x+1}\right) = \frac{1}{4}\pi\].This implies \[\frac{\frac{y}{x-1} - \frac{y}{x+1}}{1 + \frac{y^2}{(x-1)(x+1)}} = \tan\left(\frac{1}{4}\pi\right)\].
05
Simplify the Equation
Simplify the equation:\[\frac{y((x+1) - (x-1))}{(x-1)(x+1) + y^2} = 1\].Simplifying further gives:\[\frac{2y}{x^2 + 1 + y^2} = 1\].
06
Rearrange into Circle Equation
Rearrange the equation:\[2y = x^2 + y^2 + 1\].Rewrite this as:\[x^2 + (y-1)^2 = 2\].This equation represents a circle centered at \((0,1)\) with radius \(\sqrt{2}\).
07
Conclusion
The equation \(x^2 + (y-1)^2 = 2\) confirms that the locus of \(z\) is a circle of center \((0,1)\) and radius \(\sqrt{2}\), as required by the problem statement.
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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 way to visually represent complex numbers on a plane. It bears similarity to a Cartesian plane with two axes. The horizontal axis represents the real part of a complex number, while the vertical axis represents the imaginary part.
To plot a complex number like \(z = x + yi\), where \(x\) is the real part and \(y\) is the imaginary part, you find \(x\) on the horizontal axis and \(y\) on the vertical axis. The number represents a point \((x, y)\) on the plane.
To plot a complex number like \(z = x + yi\), where \(x\) is the real part and \(y\) is the imaginary part, you find \(x\) on the horizontal axis and \(y\) on the vertical axis. The number represents a point \((x, y)\) on the plane.
- The Argand Diagram is instrumental in visualizing complex numbers and their operations.
- It allows us to understand complex concepts like the argument of a complex number or transformations like rotations and reflections.
- In this exercise, the location of \(z\) on the Argand Diagram is described by a specific circle locus.
Locus
Locus refers to a set of points that satisfy a particular condition or a collection of geometrical points that obey a given rule.
In the context of complex numbers, a locus can be the path traced out by a moving point, \(z\), in the complex plane, satisfying a particular condition.
In the context of complex numbers, a locus can be the path traced out by a moving point, \(z\), in the complex plane, satisfying a particular condition.
- In this exercise, the condition is that the argument of \(\frac{z-1}{z+1}\) is \(\frac{1}{4}\pi\), which results in the locus being part of a circle.
- The concept of locus is essential in understanding how particular constraints affect the movement and location of points in the complex plane.
- This characteristic behavior forms the foundation of many geometrical interpretations of complex number operations.
Circle Equation
A circle equation is a mathematical expression representing all points equidistant from a fixed central point. The standard form of the equation for a circle in the complex plane is \((x-a)^2 + (y-b)^2 = r^2\), where \((a, b)\) is the center and \(r\) is the radius.
For example, the exercise led to the circle equation \(x^2 + (y-1)^2 = 2\).
For example, the exercise led to the circle equation \(x^2 + (y-1)^2 = 2\).
- This shows a circle centered at \((0, 1)\) with radius \(\sqrt{2}\).
- A circle in the Argand diagram indicates a set of points (complex numbers) maintaining the same angular relationship around a central point.
- Transforming equations into circle forms can aid in identifying symmetrical properties and essential relationships between points in the complex plane.
Complex Argument
The argument of a complex number is the angle parameter in its polar coordinate representation. Given a complex number \(z = a + bi\), the argument (denoted as \(\arg(z)\)) is the angle \(\theta\) formed with the positive real axis.
- The unit of the argument is typically radians when describing angles in mathematical contexts.
- The argument helps in transitioning between the Cartesian form (real and imaginary parts) and the polar form (magnitude and angle) of complex numbers.
- For the problem, having the argument of \(\frac{z-1}{z+1}\) as \(\frac{1}{4}\pi\) provides a rotational guide to solving the equation geometrically.
