I think you already have. "Determine the subset of R^2" by itself makes no sense- but I suspect the question was asking you to determine the subsets on which the equation is "hyperbolic", "elliptic", and "parabolic".

c)

My initial attempt to solve for y on this by letting u = x^2 -1 and use substitution doesnt seem to work?

Thanks

To integrate use either the trig substitution x= sec(u) or the hyperbolic substitution x= sinh(u). and so either will help you get rid of the square root.