AS/A2 Pure 4 Pure 5

P6 Topic 1: Complex numbers
Straight line loci 3: |z - z1| = |z - z2|
The locus of z when |z - z1| = |z - z2|, where z1 = x1 + iy1 and z2 = x2 + iy2 are fixed known complex numbers. 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), z1 = x1+iy1 is represented by the vector OQ where Q has coordinates (x1,y1) and z2 = x2+iy2 is represented by the vector OR where R has coordinates (x2,y2). |z - z1| represents the distance from P to Q and |z - z2| is the distance from P to R. If these two distances (shown in blue) are equal then P must be the same distance from Q as it is from R. So the locus is the perpendicular bisector of the line QR. In the diagram below you can vary the values of the complex numbers z1 and z2. Clearly, P must lie on the line shown.

Summary Cabri Geometre interactive geometry.