Yue Kwok Choy (1) It is easy to show that |z – z1| = a , where z1C, aR form a circle with centre P1(z1) and radius a , using an Argand Diagram.
(2) By putting z = x + yi and z1 = x1 + y1i , we can transform the equation to well known Cartesian form : (x – x1)2 + (y – y1)2 = a2 . The equation, in fact, is a circle with centre (x1, y1) and radius a in the rectangular plane.
(3) Squaring the equation of circle in (1), we get
|z – z1|2 = a2
We get another form of circle: , aC, cR .
Here in order not to get an imaginary or degenerate circle .
(4) Putting z = x + yi, z1 = x1 + y1i in (3) gives back the Cartesian form of the circle.
(5) Putting z = r(cos + i sin ) , z1 = (cos + i sin ) , ( , are constants) in (3) :
we get the polar form of a circle :
, with centre = (, ) and radius = .
(6) , 0 < < gives an arc and not a circle.
As in the figure, the locus gives an arc of the circle standing
we have P1P = k P2P . This then reduces to a well-known geometry problem :
The Circle of Apollonius: Given two fixed points P1 and P2, the locus of point P such that the ratio of P1P to P2P is constant , k, is a circle.
The Circle of Apollonius is not discussed here. Interested readers may consult web-sites such as:
http://jwilson.coe.uga.edu/emt725/Apollonius/Cir.html If we know that the locus is a circle, then finding the centre and radius is easier.
As in the diagram, C is the centre and AB is the diameter of the circle.
Then A and B divide P1P2 internally and externally :