Edexcel IAL - Further Pure Mathematics 2- 3.3 Loci and Regions in the Argand Diagram- Study notes  - New syllabus

Loci and Regions in the Argand Diagram

The Argand diagram represents a complex number \( z = x + iy \) as the point \( (x,y) \) in the plane.

A locus is the set of all points \( z \) that satisfy a given condition. A region is the area consisting of all points that satisfy an inequality.   

1. Locus of \( |z-a| = b \)

Let \( a = a_1 + ia_2 \). Then \( |z-a| \) represents the distance from the point \( z \) to the point \( a \).

\( |z-a| = b \)

is the set of all points at a fixed distance \( b \) from \( a \).

This is a circle with centre \( a \) and radius \( b \).

2. Locus of \( |z-a| = k|z-b| \)

This represents all points whose distances from \( a \) and \( b \) are in the constant ratio \( k \).

• If \( k = 1 \), it is the perpendicular bisector of the line joining \( a \) and \( b \).
• If \( k \ne 1 \), it is a circle (called an Apollonius circle).

3. Locus of \( \arg(z-a) = \beta \)

This represents all points \( z \) such that the line joining \( a \) to \( z \) makes a fixed angle \( \beta \) with the positive real axis.

It is a half-line (ray) starting at \( a \) and making angle \( \beta \).

4. Locus of \( \arg\!\left(\dfrac{z-a}{z-b}\right) = \beta \)

This represents all points \( z \) such that the angle between the lines from \( z \) to \( a \) and from \( z \) to \( b \) is \( \beta \).

This is a circular arc passing through \( a \) and \( b \).

Regions

(a) \( |z-a| \le b \)

This is the set of all points whose distance from \( a \) is less than or equal to \( b \).

It is the interior and boundary of the circle with centre \( a \) and radius \( b \).

(b) \( |z-a| \le |z-b| \)

This means points that are at least as close to \( a \) as to \( b \).

This is the region on one side of the perpendicular bisector of the line joining \( a \) and \( b \).