A hyperbola is one of the four types of conic sections, formed when a plane intersects both halves of a double cone. Geometrically, it is defined as the locus of all points where the absolute difference of distances to two fixed points called foci remains constant. The hyperbola consists of two separate curves called branches, which approach but never touch two straight lines called asymptotes.
The standard equations of a hyperbola depend on the orientation of its transverse axis. For a hyperbola with horizontal transverse axis, the equation is x minus h squared over a squared minus y minus k squared over b squared equals one. For a vertical transverse axis, we have y minus k squared over a squared minus x minus h squared over b squared equals one. In both cases, h and k represent the coordinates of the center.
A hyperbola has several key components. The foci are two fixed points that define the hyperbola. The vertices are where the hyperbola intersects its transverse axis. The center is the midpoint between the vertices. The transverse axis has length two a, while the conjugate axis has length two b. The asymptotes are lines that the hyperbola approaches but never touches. These components are related by the equation c squared equals a squared plus b squared.
The fundamental property of a hyperbola is that for any point P on the curve, the absolute difference of distances from P to the two foci equals two a, which is constant. Let's watch as point P moves along the hyperbola. Notice how the lines to the foci change length, but their difference remains constant at two a. This constant difference property is what defines the hyperbola and distinguishes it from other conic sections.
To summarize what we have learned about hyperbolas: A hyperbola is a conic section consisting of two separate branches. It is defined as the locus of points where the absolute difference of distances to two foci remains constant. The standard equations vary based on whether the transverse axis is horizontal or vertical. Key components include the foci, vertices, center, and asymptotes. Hyperbolas have important applications in physics, engineering, and navigation systems.