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.

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.

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## 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.

## This question appears in the following syllabi:

Syllabus | Module | Section | Topic | Exam Year |
---|---|---|---|---|

AQA A-Level (UK - Pre-2017) | FP2 | Complex Numbers | Loci | - |

AQA AS Further Maths 2017 | Pure Maths | Complex Numbers and Geometry | Loci | - |

AQA AS/A2 Further Maths 2017 | Pure Maths | Complex Numbers and Geometry | Loci | - |

CCEA A-Level (NI) | FP1 | Complex Numbers | Loci | - |

CIE A-Level (UK) | P3 | Complex Numbers | Loci | - |

Edexcel A-Level (UK - Pre-2017) | FP2 | Complex Numbers | Loci | - |

Edexcel AS Further Maths 2017 | Core Pure Maths | Argand Diagrams | Loci | - |

Edexcel AS/A2 Further Maths 2017 | Core Pure Maths | Argand Diagrams | Loci | - |

Methods (UK) | M3 | Complex Numbers | Loci | - |

OCR A-Level (UK - Pre-2017) | FP1 | Complex Numbers | Loci | - |

OCR AS Further Maths 2017 | Pure Core | Argand Diagrams and Loci | Loci | - |

OCR MEI AS Further Maths 2017 | Core Pure A | Argand Diagrams and Loci | Loci | - |

OCR-MEI A-Level (UK - Pre-2017) | FP1 | Complex Numbers | Loci | - |

Scottish Advanced Highers | M2 | Complex Numbers | Loci | - |

Scottish (Highers + Advanced) | AM2 | Complex Numbers | Loci | - |

Universal (all site questions) | C | Complex Numbers | Loci | - |

WJEC A-Level (Wales) | FP1 | Complex Numbers | Loci | - |