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.
$\endgroup$3 Answers
$\begingroup$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.
$\endgroup$ $\begingroup$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.
$\endgroup$ $\begingroup$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.
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