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What is a Hyperbola?

By P.S. Jones
Updated May 21, 2024
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A hyperbola is mathematical term for a curve on a plane that has two branches that are the mirror images of each other. Like the similar parabola, the hyperbola is an open curve that has no ending. This means that it in theory it will go on infinitely, unlike the circle or the ellipse.

This should not be confused with the literary term hyperbole. Both terms comes from a Greek word that translates to “over-thrown” or “excessive”. However, hyperbole is a literary concept that describes a statement that is greatly exaggerated for emphasis. It is most common seen in poetry or casual speech. The term hyperbola is generally thought to be coined by Apollonius of Perga in his work with conics.

Cones have four curves called conics, which include hyperbolas and parabolas, as well as the circles and the ellipses. Each section is defined by its eccentricity, or by how much it deviates from being a circle. For example, the eccentricity of a circle is zero. A hyperbola’s eccentricity is greater than one and parabola’s eccentricity is less than one. On the other hand, the eccentricity of an ellipse is less than one but more than zero.

A hyperbola has several characteristics to it. It has two focal points, which may also be called foci. These two points are connected by a line called the transverse axis, and that line’s midpoint marks the hyperbola’s center. Further, the line that is perpendicular to the transverse axis is called the conjugate axis. Together the conjugate axis and the transverse axis make up the two principal axis of the hyperbola. These two axis are important, because a parabola must be symmetrical across both of these lines.

Hyperbolas have applications outside of the theoretical world. Take for example a ripple of water that forms concentric circles. As those circles intersect, they form hyperbolas. Both sound and light waves mimics this behavior. Radar is one particular area of technology that uses the hyperbola in its scientific reasoning.

Hyperbolas can be found in space, too. Orbiting planets or moons follow an elliptical orbital path. However, any object that passes through a solar system and does not orbit will follow a hyperbolic path. A comet is an example of a hyperbolic path through space.

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