A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane.
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An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant.
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A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant.
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