Eliminating Variables in Circles Equations

[zpconn]

Homework Statement

I have the equations x = a * sinh(u) / (cos(v) + cosh(u)) and y = a * sin(v) / (cos(v) + cosh(u)). I believe that if a value of u is fixed, then the locus of (x,y) satisfying these equations is a circle depending on v. Similarly, I believe that if a value of v is fixed, then the locus of (x,y) satisfying these equations is a circle depending on u.

I would like to verify this.

Homework Equations

The Attempt at a Solution

It seems the easiest way is to find the equations of the circles by somehow eliminating either u or v, but I can't figure out how. I've employed various identities, such as cosh^2 - sinh^2 = 1, but I can never fully eliminate one variable.

 

[zpconn]

The coordinate system I'm working with is slightly different, but my conjecture about the level curves for u = const. and v = const. is based off this Wikipedia article: http://en.wikipedia.org/wiki/Bipolar_coordinates

I cannot figure out, however, how Wikipedia managed to derive the equations in the section "Curves of constant sigma and tau." I suspect a similar method will work in my case.