Final answer:
The general forms used by an ellipse or hyperbola come from the equation of a conic section, and these shapes play a significant role in the motion of celestial bodies according to Kepler's laws.
Step-by-step explanation:
The general form used by an ellipse or hyperbola, which are both conic sections, can be derived from the equation of a conic section. All conic sections, which include the circle, ellipse, parabola, and hyperbola, are formed by the intersection of a plane with a cone. An ellipse, specifically, is a closed curve where the sum of the distances from any point on the curve to the two foci is a constant. A hyperbola, by contrast, is an open curve with two separate branches, and it also has a specific general form stemming from the conic section equation.
Additionally, the properties of elliptical orbits are described by Kepler's laws, which state that planets move in ellipses with the Sun at one focus. This understanding is critical in physics and astronomy for describing the motion of celestial bodies. Similarly, the trajectories of objects in unbound gravitational interactions are represented by parabolas or hyperbolas, again conic sections, confirming the importance of these shapes in various fields of science.