a) I am unsure how to show that the point MUST lie on the curve. All I can think of doing is finding a point that I know is equidistant between the curve and point, and then substituting these values into the equation of the curve? i.e. (1.5,0). Is this the correct way to approach this part of the problem.

To show that the curve is a hyperbola, do I just need to rearrange the equation they have given me into the form of a hyperbola ie. (x^2/a^2 - y^2/b^2 = 1) If so I have done this, and I have an equation that resembles a hyperbola, but I'm not sure this really SHOWS that the curve is a hyperbola. I think maybe I am missing something, any help would be great!

yep, just rearrange to show that it is a hyperbola.

as for the other questions, umm... I'll think about it