AS/A2 Pure 4 Pure 5

P6 Topic 1: Complex numbers
Arc loci 1: arg (z-z1)/(z-z2) = q
The locus of z when arg (z-z1)/(z-z2) = q, where z1 = x1 + iy1 and z2 = x2 + iy2 are fixed known complex numbers and q is a fixed known angle. 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 vector from P to Q and z - z2 is the vector from P to R. The locus equation requires that the angle between the directions of these vectors is q. The diagram shows that (due to the "angle in same segment" circle theorem) this locus is in fact an arc from Q to R. In the diagram below you can vary the values of the complex numbers z1 and z2 and the value q.

Summary In the special case when q = p/2, the arc becomes a semicircle with centre at the midpoint of QR. Cabri Geometre interactive geometry.