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polynomials - Eliminating variables from system of equation using Eliminate or Solve


I have a system of equations for algebraic curve given by the zero locus of some polynomial encoded in the system of equations (I want to eliminate variable z and get algebraic curve in terms of x and y, C12 and C22 can be assumed to be integers and even positive if that helps):


Eliminate[{y == z^(2 + 2 C12)/(-x^2 + z^2), 
1 == ((x - z) (x/z)^(2 C12) z^(-2 - 2 C12 + 2 C22) (x + z))/(-1 + 
z^2)}, {z}]

Eliminate::ifun: Inverse functions are being used by Eliminate, so some solutions may not be found; use Reduce for complete solution information.


However Mathematica can't eliminate z.


When I use Solve I get:


Solve::nsmet: This system cannot be solved with the methods available to Solve.



Maple gives some answers but they aren't very useful, for instance they don't give algebraic curve that I want.


Is there any smart way to do what I want?




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