Loci in Argand Diagrams (Edexcel A-Level Further Mathematics): Revision Notes

1.1.5 Loci in Argand Diagrams

Overview

The locus of a complex number represents a set of points that satisfy specific geometric conditions on an Argand diagram. Argand diagrams allow complex numbers to be visualised with their real parts on the $x-axis$ and imaginary parts on the $y-axis$.

Argand Diagram:

Common Loci Forms

Locus of the Form $|z - z_1| = r$ (Circle)

This represents a circle centred at $z_1$ with radius $r$.

• Equation:$|z - z_1| = r$
• Geometrical Interpretation: Set of points $z$ at a constant distance $r$ from $z_1$. Diagram:

Locus of the Form $|z - z_1| = |z - z_2|$ (Perpendicular Bisector)

This represents the perpendicular bisector of the line segment joining two fixed points $z_1$ and $z_2$.

• Equation: $|z - z_1| = |z - z_2|$
• Geometrical Interpretation: Set of points equidistant from $z_1$ and $z_2$.

Locus of the Form $|z - z_1| = k|z - z_2|$ (Circle or Line)

• If $k = 1$: The locus is the perpendicular bisector of $z_1$ and $z_2$.
• If $k \neq 1$: The locus is a circle.

Locus of the Form $\arg(z - z_1) = \theta$ (Ray)

This represents a half-line starting from $z_1$ at an angle $\theta$ from the positive real axis.

• Equation: $\arg(z - z_1) = \theta$
• Geometrical Interpretation: Ray starting at $z_1$ extending in the direction of $\theta$.

Worked Example

Note Summary

Common Mistakes:

1. Incorrect Interpretation of Quadrants: Errors in locating points on the Argand diagram.
2. Confusing Modulus with Argument: Mistaking $|z|$ for $\arg(z)$
3. Misapplication of Distance Ratios: Failing to correctly identify loci as circles or lines based on ratio conditions.
4. Forgetting to Adjust Arguments: Not properly handling angles in different quadrants.
5. Neglecting Graphical Accuracy: Inaccurate sketches of loci on Argand diagrams.

Key Formulas:

Circle:

$$|z - z_1| = r$$

(Centre: $z_1$, Radius: $r$)

Perpendicular Bisector:

$$|z - z_1| = |z - z_2|$$

(Locus: Equidistant from $z_1$ and $z_2$)

Ratio of Distances:

$$|z - z_1| = k|z - z_2|$$

($k = 1$: Line, $k \neq 1$: Circle)

Ray:

$$\arg(z - z_1) = \theta$$

(Ray from $z_1$ at angle $\theta$)

General Modulus:

$$|z| = \sqrt{x^2 + y^2}$$