Where can hyperbolas be found?

Answer 1

A hyperbola is a conic section. Therefore, it can be produced by slicing a double cone. Half of a hyperbola, just one of its two branches, can be found by slicing a single cone. The cone must be sliced by a plane that is angled sufficiently so that it would intersect a double cone twice.

This suggests a way to form one. If one points a flashlight directly at a wall, one sees a circle; moving back increases the radius of the circle. This indicates that light emerges from the flashlight in a cone shape, with its apex at the light bulb. If the flashlight is tilted, the shape of the spot of light on the wall elongates, first becoming an ellipse, then a parabola. Tilting further yields a branch of a hyperbola, as the cone is now inclined in such a way that the plane (the wall) intersects the hypothetical double cone twice.

In celestial mechanics, a body that passes by a more massive body, entering its gravitational field, may, if it has sufficient energy, "slingshot" around it instead of becoming trapped in orbit or colliding. If it has exactly enough energy to do this, its trajectory will be a parabola with the massive body at the focus; if it has any more energy, its trajectory will be half of a hyperbola, with the massive body at one of the foci. The reasoning behind this is not nearly so simple as with the flashlight, however.