AS/A2 Pure 4 Pure 5

P6 Topic 1: Complex numbers
Circle loci 1: |z| = a
The locus of z when |z| = a, where a is a fixed constant. If the complex number z=x+iy is represented on an Argand diagram by the vector going from the origin to the point P with coordinates (x,y), then the equation |z| = a requires the distance of P from the origin, ie., the vector OP, to be of length a units. So the locus is a circle, centre the origin, with radius a units. In the diagram below you can vary the value of a. Clearly, P must lie on the circle shown.

Summary |z| is the modulus of z. If z is represented on an Argand diagram by the vector OP then |z| is the length of the vector OP. Cabri Geometre interactive geometry.