AS/A2 Pure 4 Pure 5

P6 Topic 1: Complex numbers
Straight line loci 2: arg (z - z1) = q
The locus of z when arg (z - z1) = q, where q is a fixed known angle and z1 = x1 + iy1 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 z1 = x1+iy1 is a known complex number represented by the vector OQ where Q has coordinates (x1,y1), then the equation arg (z - z1) = q requires the vector from Q to P to make an angle of q with the positive x axis (the real axis). So the locus is a line segment from Q at an angle q to the x-axis. In the diagram below you can vary the value of q and the complex number z1. Clearly, P must lie on the half-line shown.

Summary Cabri Geometre interactive geometry.