I have just posted an experimental answer checker to the CVS repository
in pg/macros/parserImplicitEquation.pl that should handle the situation
you are interested in. It attempts to compare two implicitly defined
equations to see if they have the same solutions. The check is not
perfect, but it might be worth trying out.
Read the comments at the top of the file for some caveats about what
can go wrong. Because of the algorithm used to check the answers, there
are always ways for the student to write correct answers that will not
be marked correct, and incorrect answers that will pass. For example,
if f(x,y) = 0, then for large n the algorithm will mark (f(x,y))^n = 0
as incorrect, and it will not be able to locate the solutions for even
n at all (the algorithm needs sign changes to identify solutions). It
will also mark (x^2+y^2-.01)f(x,y)=0 as correct, since the algorithm is
unlikely to notice the additional zeros introduced by the small circle.
But it WILL recognize xy=4, x/2=2/y, x=4/y, x-4/y=0, (x^2+y^2)(xy-4)=0,
and so on as being the same thing. It will also get x^2-y^2=0 as
|x|=|y|, and x^2-y^2=1 as |y|=sqrt(x^2-1). So it has some promise.
Try it out and see what you think. If you DO use it, you might want to
view past answers on the problems that you use it on, and see whether
it is marking things wrong.
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