Although both figures are conic sections, parabolas and hyperbolas are generated through different methods. A parabola is formed when a plane parallel to a cone's side cuts through the cone. A hyperbola results from the intersection of the plane and the cone, but with the plane at an orientation that is not parallel to the side of the cone.
The parabola and the hyperbola also differ in terms of their properties as conic sections. Any conic section may be defined as a path or locus of points at a particular distance from a fixed line and a fixed point. These two types of conic sections can be differentiated by calculating the eccentricity, which is equal to the distance of the point on the curve to the fixed point divided by the perpendicular distance of that same point on the curve to the fixed line. A parabola has an eccentricity value of one, whereas a hyperbola has an eccentricity greater than one.
The graphs of these two curves are also slightly different. Hyperbolas open more widely than parabolas. The more noticeable difference in their graphs is that a hyperbola has two curves that mirror each other and open in opposing sides. On the other hand, a parabola has only one curve.