The elements of a hyperbola: foci, transverse axis, conjugate axis, vertices, centre, etc. | 3

Описание: A hyperbola is the set of all points in a plane such that, for any point P in the set, the absolute value of the difference of the distances to two fixed points, called the foci, is constant - specifically 2a, the length of the transverse axis of the hyperbola. Like circles, ellipses and parabolas, hyperbolas are conic sections - in other words, curves obtained by slicing a right circular cone with a plane. Specifically, we obtain a hyperbola when the plane is parallel to the axis of rotation of the double cone and perpendicularly intersects its two bases.

The foci of a hyperbola are two fixed points in the plane around which the hyperbola is constructed. The closer the foci are to the vertices, the narrower the opening of the two branches of the hyperbola; conversely, the farther the foci are from the vertices, the wider the opening of the two branches of the hyperbola and the straighter the two branches. Hyperbolas and ellipses have two foci, unlike parabolas, which have only one focus, and unlike circles, which have no foci.

The transverse axis and the conjugate axis of a hyperbola are its two axes of symmetry, which intersect perpendicularly at the centre of the hyperbola. They define the shape and size of the hyperbola. For instance, the longer the transverse axis, the farther apart the two branches become. In a hyperbola, the transverse axis can be longer than, shorter than, or equal to the conjugate axis. The longer the transverse axis is compared to the conjugate axis, the narrower the opening of the two branches of the hyperbola; conversely, the shorter the transverse axis is compared to the conjugate axis, the wider the opening of the two branches of the hyperbola and the straighter the two branches. If the transverse axis and the conjugate axis are equal in length, the hyperbola becomes a rectangular hyperbola, and its two asymptotes are perpendicular.

The vertices of a hyperbola are the two points at which the hyperbola intersects its transverse axis, reaches its minimum distance from the centre, and turns around. If the hyperbola opens horizontally - as is the case in this diagram - the coordinates of the two vertices are (-a, 0) and (a, 0), respectively, where a is the length of the semi-transverse axis. The two vertices of a hyperbola are the endpoints of its transverse axis. Thus, the distance between the two vertices is equal to the length of the transverse axis, which is 2a.

The endpoints of the conjugate axis don’t lie on the hyperbola. The conjugate axis never intersects the hyperbola. As a result, hyperbolas have no co-vertices, unlike ellipses, which have two co-vertices because their minor axis intersects them.

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