
A hyperbola is a conic section formed by intersecting a right circular conic surface and a plane that intersects both halves of the cone. The equations most often used for a hyperbola are: where a is the semimajor axis parallel to the xaxis, b is the semiminor axis parallel to the yaxis, and the point (h,k) is the center of the hyperbola, and where a is the semiminor axis parallel to the xaxis, b is the semimajor axis parallel to the yaxis, and the point (h,k) is the center of the hyperbola. 

A hyperbola can be defined as a set of points where the ratio of the distance from a fixed line called the directrix and the distance from the point is equal to the ratio of c to a. The equation for the directrices of a hyperbola with an east/west opening is .
The equation for the directrices of a hyperbola with a north/south opening is
.
The foci for a hyperbola are the points at for a hyperbola with an east/west opening and for a hyperbola with a north/south opening. In manipulative 1, click on the check box marked 'Show directrices'. Click on the blue point on the hyperbola and drag it. Notice that the ratio remains constant for a particular hyperbola. 
The eccentricity of a hyperbola can be considered as how far the hyperbola deviates from a circle. The larger the eccentricity, the flatter the hyperbolic curve. The formula for the eccentricity of a hyperbolic curve is
A flashlight makes a cone of light. If a flashlight is held close to a wall and parallel to the wall, the plane of the wall 'cuts' the cone at in such a way that the 'edges' of the light form a hyperbola.  
 
The path of an object in space relative to a much larger object can be a hyperbola. The path of the object is a hyperbolic trajectory if the speed of the smaller object relative to the larger object is more than escape velocity. In actuality, very few hyperbolic trajectories are observed in nature. This is because an object on a hyperbolic trajectory 'breaks free' of the gravitational field of the larger object (in this case the sun). Because it breaks free, the object does not return close to the larger object so is rarely observed. 

A hyperboloid is a 3dimensional figure created by rotating a hyperbola about a line. Figures 4 and 5 show two types of hyperboloids. 

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