If you are given a curve and you are told that it is either a parabola or a hyperbola, how can you tell definitively which one it is? No coordinates provided.
Geometrical method preferred, i.e. visual inspection, drawing lines, or comparing distances, but not curve fitting.
Look to see whether it has asymptotes, i.e., whether the "tails" of the curve approach straight lines or not. If so: hyperbola; if not, parabola.
A parabola is the limiting curve of a sequence of ever narrower hyperbolas (with one focus converging to the parabola's focus and the other focus escaping to infinity).
Thus any finite segement of a parabola can be approximated uniformly by hyperbolas, so simply looking at a curve you can never be sure that it's actually a parabola rather than a narrow hyperbola.
If it is clear from what you see of the curve that its two ends have asymptotes in different directions, then you can be sure it's not a parabola, though.
You cam make a geometrical construction of rays parallel to the parabola/hyperbola axis reflecting on the concave surface and see if they intersect at a single point, in which case it would be a parabola.
