The following problem was posed as Dan Pedoe's Exercise 0.9:
If (z-a)/(z-b) = r(cos(θ) + i sin(θ), show that the curves r = constant, and θ = constant, for fixed complex numbers a and b and varying Z, are circles of orthogonal coaxal systems, the one which has the points a and b as limiting points, and the other of circles which pass through these two points.
Since this problem occurred in an introductory chapter of the text, my expectation was that it was going to be a straightforward application of his quick review on complex numbers. It proved to be something more interesting.
In fact, establishing the part when r is constant is pretty straightforward once you recognize that the set of points whose distance to two fixed points is a constant ratio r is in fact a circle.
But the case for fixed θ proved the more interesting/complicated.
I recommend that anyone who has read this much should try to work it out before reading the solution linked below. I would love to see another approach to the solution if you find one.
For anyone who is interested, here is a link to a paper which provides what I hope will be a pretty self-contained solution intended for a reader who has studied senior high school level math. It includes a little background material for anyone who might be a little rusty.
Note: One point not explicitly mentioned in the paper is that when θ is held constant and r varies, θ presents itself as the angle between the perpendicular bisector of the segment ab and the line joining the center of the so-called θ-circle and the point a (or b). Thus the changing θ-circles can be visualized for θ in [0,π/2].
