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 visualized 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 centered 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:

lightbulbExample

Example Find the locus of $|z - (3 + 4i)| = 5$

Step 1: Identify center and radius.

z_1 = 3 + 4i, \quad r = 5

Step 2: Plot the circle with center $(3, 4)$ and radius $5$ .

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 $z_2$ .

infoNote

Example Solve $|z - (1 + i)| = |z - (3 + 5i)|$

Step 1: Identify $z_1 = 1 + i$ and $z_2 = 3 + 5i$

Step 2: This is the perpendicular bisector of the line joining $(1, 1)$ and $(3, 5)$

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.

lightbulbExample

Example Solve $|z - (2 + i)| = 2|z - (4 + 3i)|$

Step 1: Recognize the circle condition, since $k = 2 \neq 1$

Step 2: Plot the circle based on geometric relationships.

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 $\theta$ .

lightbulbExample

Example Find the locus of $\arg(z - (1 + 2i)) = \frac{\pi}{4}$

Step 1: Identify $z_1 = 1 + 2i$

Step 2: Plot a ray starting at $(1, 2)$ making an angle of $\frac{\pi}{4}$ with the real axis.

Worked Example

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Example: Find the locus represented by $|z - (2 - i)| = 3$

Step 1: Identify center and radius.

z_1 = 2 - i, \quad r = 3

Step 2: The locus is a circle centered at $(2, -1)$ with radius $3$ .

Step 3: Plot the circle on the Argand diagram.

Note Summary

Common Mistakes:

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

Key Formulas:

Circle:

|z - z_1| = r

(Center: $z_1$ , 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}

Loci in Argand Diagrams Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

1.1.5 Loci in Argand Diagrams

Overview

Common Loci Forms

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

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

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

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

Worked Example

Note Summary

Common Mistakes:

Key Formulas:

500K+ Students Use These Powerful Tools to Master Loci in Argand Diagrams For their A-Level Exams.

Other Revision Notes related to Loci in Argand Diagrams you should explore

Loci in Argand Diagrams Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

1.1.5 Loci in Argand Diagrams

Overview

Common Loci Forms

Locus of the Form ∣z−z1∣=r|z - z_1| = r∣z−z1​∣=r (Circle)

Locus of the Form ∣z−z1∣=∣z−z2∣|z - z_1| = |z - z_2|∣z−z1​∣=∣z−z2​∣ (Perpendicular Bisector)

Locus of the Form ∣z−z1∣=k∣z−z2∣|z - z_1| = k|z - z_2|∣z−z1​∣=k∣z−z2​∣ (Circle or Line)

Locus of the Form arg⁡(z−z1)=θ\arg(z - z_1) = \thetaarg(z−z1​)=θ (Ray)

Worked Example

Note Summary

Common Mistakes:

Key Formulas:

500K+ Students Use These Powerful Tools to Master Loci in Argand Diagrams For their A-Level Exams.

Other Revision Notes related to Loci in Argand Diagrams you should explore

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

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

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

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