# Loci

Loci in Complex Numbers

Concept of Loci

• Understand that a locus is a set of points satisfying a certain condition.
• Recognize that in complex numbers, the locus is often represented on the Argand diagram, a plane with real and imaginary axes.
• Append that the loci in complex numbers can represent an array of shapes such as lines, circles, or segments of these.

Argand Diagram

• Appreciate the significance of the Argand diagram for visualizing complex numbers, where the horizontal axis represents the real part, and the vertical axis represents the imaginary part.
• Associate that in the Argand diagram, a complex number z = a + bi corresponds to a point (a, b).

Forms of Loci

•  Be familiar with the circle locus, which is in the form ** z - a = r, where **z is a variable complex number, a is a set complex number, and r is the radius. This represents a circle with the centre at a.
• Grasp the line (or radius) locus, written as arg(z - a) = θ. This represents a half-line from the point a forming an angle θ with the positive real axis in the anti-clockwise direction.
• Understand the semi-circle locus which is represented as arg(a - z) - arg(z - b) = π/2. This locus forms a semicircle with a and b being the endpoints of the diameter.

Solving Loci Problems

• Become well-versed with the process of visualizing the loci first on the Argand diagram before proceeding to solve them algebraically.
• Remember that when you have two loci conditions, the intersection points represent solutions satisfying both conditions.
• In cases where a condition includes equality or inequality signs, realise that the solution can either be on the locus, inside it or outside it based on the context.

Loci Transformations

• Master the transformations such as translation, rotation, and scale can be applied on loci. These transformations result in the motion of the entire set of points that form the locus.
• For example, z + a represents a translation of the locus of z by -a.

Revise these points critically; applying loci concepts to complex numbers is vital for understanding more complicated relationships between complex numbers.