If z=x+ħy, determine the equations of the two loci: (a) |(z+2)/(z)|=3 and (b) {(z+2)/(z)}=(π)/(4

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If z=x+ħy, determine the equations of the two loci: (a)...

Key Concepts

Complex Number Representation

Complex numbers are written in the form z = x + iy (or z = x + ?y in some notations), where 'x' represents the real part and 'y' represents the imaginary part. This representation allows us to work with two-dimensional numbers and perform algebraic operations, making it foundational in understanding behavior in the complex plane.

Modulus of a Complex Number

The modulus of a complex number, denoted |z|, represents its distance from the origin in the complex plane. It is defined by the formula |z| = ?(x² + y²) for a complex number z = x + iy. This concept is critical when equating the magnitudes in rational expressions involving complex numbers.

Argument of a Complex Number

The argument of a complex number, often noted as arg(z), is the angle formed by the line representing the complex number and the positive real axis. It is essential for understanding rotations and orientations in the complex plane and is particularly important when dealing with equations that involve the phase relationship between complex numbers.

Locus of Points in the Complex Plane

A locus in the complex plane is the set of all points (complex numbers) that satisfy a given condition. In problems involving complex equations, either in terms of modulus or argument, determining the locus is key to understanding the geometric nature of the solutions, which can be circles, lines, or more complex curves.

Complex Function Transformations

Transformations of complex numbers, such as the mapping given by w = (z+2)/z, are functions that reassign a complex number z to another complex number w. Studying these mappings involves analyzing how they affect the modulus and argument of z, which is crucial for solving equations that set conditions on these properties.

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