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The locus of z when |z-z_1| < a, where a is a fixed constant and z_1 is a fixed known complex number.
If z=x+iy is represented on an Argand diagram by the vector going from the origin to the point P with coordinates (x,y), and z_1 = x_1+iy_1 is a known complex number represented by the vector OQ where Q has coordinates (x_1,y_1), then the equation |z - z_1| < a requires the distance of P from Q to be at most a units. So the locus is the interior of a circle, centre (x_1,y_1), with radius a units. In the diagram below you can vary the value of a and the vector z_1. Clearly, P must lie within the circle shown.

## Glossary

### argand diagram

A geometrical presentation of a complex number; using a real and imaginary axis to plot a+ib as (a,b).

### circle

a conic curve with equation (x-a)²+(y-b)²=r²

### complex number

A number of the form a+bi where i is the square root of -1, and a and b are real.

### constant

a value in a formula or equation that cannot change.

### equation

A statement that two mathematical expressions are equal.

### locus

A set of points that fit a given rule.

### origin

The point from where all measurements of coordinates are made; usually the point where the two axes of a graph cross.

### vector

A mathematical object with magnitude and direction.

Full Glossary List

## This question appears in the following syllabi:

SyllabusModuleSectionTopicExam Year
AQA A-Level (UK - Pre-2017)FP2Complex NumbersLoci-
AQA AS Further Maths 2017Pure MathsComplex Numbers and GeometryLoci-
AQA AS/A2 Further Maths 2017Pure MathsComplex Numbers and GeometryLoci-
CCEA A-Level (NI)FP1Complex NumbersLoci-
CIE A-Level (UK)P3Complex NumbersLoci-
Edexcel A-Level (UK - Pre-2017)FP2Complex NumbersLoci-
Edexcel AS Further Maths 2017Core Pure MathsArgand DiagramsLoci-
Edexcel AS/A2 Further Maths 2017Core Pure MathsArgand DiagramsLoci-
Methods (UK)M3Complex NumbersLoci-
OCR A-Level (UK - Pre-2017)FP1Complex NumbersLoci-
OCR AS Further Maths 2017Pure CoreArgand Diagrams and LociLoci-
OCR MEI AS Further Maths 2017Core Pure AArgand Diagrams and LociLoci-
OCR-MEI A-Level (UK - Pre-2017)FP1Complex NumbersLoci-