How many different equations of a circle are there?

Question

$$\frac{1+x^2-y^2}{(1+x^2-y^2)^2+4x^2y^2}=cst$$ reduces to the equation of a circle: $x^2+y^2=2$

How many other equations reduce to circles?

Answer 1

There are infinitely many ways for equations to be reducible (what ever that really is) to the equation of a circle. You circle is not a circle it is the specific case of the unit circle at the origin.