Q:

# What Are the Foci of a Hyperbola?

A:

The foci of a hyperbola are points located inside the curve of each branch of the hyperbola that are equidistant from the center point on the transverse axis. For any point on a hyperbola, the difference between its distance from one focus and its distance from the other focus is always a fixed amount.

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Credit: Wesley Fryer CC-BY-2.0

A hyperbola is a curve made up of two identical parabolas, called "branches," which curve away from a center point. Each branch has a vertex, the point on the branch closest to the center, and a focus, which is located farther from the center inside the curve of the branch. The distance from each vertex to the center is designated as "a," and the distance from each focus to the center is designated "c." For any given hyperbola, c is greater than a and both values are fixed. The foci and vertices of a hyperbola are also proportional to the distance between the center and one end of the conjugate axis, which is designated as "b." The conjugate axis is perpendicular to the transverse axis. The relationship between a, b, and c in a hyperbola is represented by the equation, "a squared plus b squared equals c squared." While this equation looks identical to the Pythagorean theorem, the values involved have a completely separate meaning.