Exam Questions - Loci in the complex plane

Understanding the Loci in the Complex Plane

The locus is a path or set of points satisfying specific conditions. In terms of the complex plane, these conditions are often related to complex numbers.
The complex plane is a two-dimensional plane where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers.
In this scenario, each complex number is represented by a point in the plane, using the real part as the x-coordinate and the imaginary part as the y-coordinate.

Basics of Loci

Loci can be represented through equations like **

z - a

= r**, where ‘a’ is a complex number and ‘r’ is a positive real number. This represents a circle where ‘a’ is the centre and ‘r’ is the radius.

The equations **

z - a

z - b

** or arg(z - a) = θ are also commonly used to represent different kinds of loci in the complex plane. The first equation represents the locus of points that are equidistant from two fixed points ‘a’ and ‘b’, effectively representing a perpendicular bisector. The second equation represents an argument or a ray from the point ‘a’ at an angle ‘θ’ anti-clockwise from the real axis.

Remember, ‘arg’ denotes the argument function, defining the angle made with the positive real axis.

Transformations of the Complex Plane

It’s important to understand general transformations (translation, rotation, dilation) as the locus within the complex plane can be transformed similarly.
If ‘w = az + b’ is a transformation, where a and b are complex constants and z and w are complex variables, understanding transformations’ effects on the shape and position of loci are key.
To visualise this, begin by considering simpler transformations such as translations that shift the locus without altering its shape.

Complex Roots and Their Symmetry

Some questions might ask to show the roots of a polynomial equation with complex coefficients.
The roots of these equations can be represented as points in the complex plane, which exhibit patterns of symmetry owing to the conjugate root theorem (if you have a complex number as a root, then its conjugate is a root too), forming distinct loci.

Working with Loci Questions

Thoroughly read each question and fully understand what is asked before attempting to sketch loci.
Carefully interpret given conditions to accurately express and represent the locus in the complex plane.
For the transformation-based problems, develop a systematic approach - identify type of transformation, apply it to a suitable loci representation and then sketch transformed loci.
Practice is key: Work through past paper problems to gain familiarity with common patterns and marker expectations.

Real Life Connections

The concept of loci in complex plane finds usage in physics, where it helps in understanding quantum mechanics, wave physics and vibrations.
It is also used in engineering fields to analyse system stability and understand frequency response.