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A hyperbolic comet is one with an orbit that forms a hyperbola rather than the usual ellipse. Its orbital eccentricity is greater than 1. A comet with an eccentricity of 1 is a parabolic comet, and if just under 1, a near-parabolic comet or near-hyperbolic comet. One criteria for near-parabolic is between 0.99 and 1.
When speaking of hyperbolic comets, often all of these are in the discussion. A few thousand comets with an eccentricity close to 1 are known. Some may have come from beyond the solar system (i.e., interstellar objects that are comets), but that is not clear: an orbit from the furthest reaches of the Oort Cloud is naturally extremely close to parabolic, and it is presumed possible that speed could be increased, such as through an interaction with another celestial object. It is also unclear whether they will travel beyond the solar system: such an interaction may well slow them on their way out and they are passing through the interplanetary medium and interstellar medium. The recently-spotted interstellar objects had eccentricities not as close to 1: the first interstellar interloper had an eccentricity of 1.2 (and the subsequent two even higher), whereas the comets not declared to be interstellar have had eccentricities less than 1.1, with only a handful above 1.01.