AS/A2 Pure 4 Pure 5

P6 Topic 1: Complex numbers
Straight line loci 4: |z - z1| = a|z - z2|
The locus of z when |z - z1| = a|z - z2|, where z1 = x1 + iy1 and z2 = x2 + iy2 are fixed known complex numbers and a is a fixed known constant. 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. The locus equation requires that PQ is equal to a times PR. The shape of this locus is not obvious, and you should use algebra at this point to find the equation of the locus, which will turn out to be a circle. In the diagram below you can vary the values of the complex numbers z1 and z2 and the value a.

Summary Note that the above display does not work for all values of a. You need to keep a close to 2. Cabri Geometre interactive geometry.