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A hyperbola and its conjugate hyperbola. In geometry, a conjugate hyperbola to a given hyperbola shares the same asymptotes but lies in the opposite two sectors of the plane compared to the original hyperbola. A hyperbola and its conjugate may be constructed as conic sections obtained from an intersecting plane that meets tangent double cones ...
Since the eccentricity of a hyperbola is always greater than one, ... change the sign on the right to obtain the equation of the conjugate hyperbola: ...
The eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axis a: that is, = (lacking a center, the linear eccentricity for parabolas is not defined). It is worth to note that a parabola can be treated as an ellipse or a hyperbola, but with one focal point at infinity .
When the conjugate of the unit hyperbola is in use, the alternative radial length is =. The unit hyperbola is a special case of the rectangular hyperbola, with a particular orientation, location, and scale. As such, its eccentricity equals . [1]
A pencil of confocal ellipses and hyperbolas is specified by choice of linear eccentricity c (the x-coordinate of one focus) and can be parametrized by the semi-major axis a (the x-coordinate of the intersection of a specific conic in the pencil and the x-axis). When 0 < a < c the conic is a hyperbola; when c < a the conic is an ellipse.
(Given the lunar orbit's eccentricity e = 0.0549, its semi-minor axis is 383,800 km. Thus the Moon's orbit is almost circular.) Thus the Moon's orbit is almost circular.) The barycentric lunar orbit, on the other hand, has a semi-major axis of 379,730 km, the Earth's counter-orbit taking up the difference, 4,670 km.
With eccentricity just over 1 the hyperbola is a sharp "v" shape. At = the asymptotes are at right angles. With > the asymptotes are more than 120° apart, and the periapsis distance is greater than the semi major axis. As eccentricity increases further the motion approaches a straight line.
A ray through the unit hyperbola x 2 − y 2 = 1 at the point (cosh a, sinh a), where a is twice the area between the ray, the hyperbola, and the x-axis. For points on the hyperbola below the x-axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions).