Image Comets travel in either elliptical, parabolic, or hyperbolic orbits.

Image The designer of the McDonnell Planetarium in St. Louis, Gyo Obata, was inspired to use the hyperbola by the fact that certain comets travel in hyperbolic orbits.

Conic Sections: Hyperbola

If a plane cuts a double-cone through both cones, the resulting intersection is a hyperbola. Because a hyperbola was originally obtained by slicing a cone (a double-cone) with a plane, it is also classified in the category of conic sections. 		Image

Image As for the ellipse, the direct definition of the hyperbola also requires the selection of two fixed points F and G, called foci, and one constant quantity "c". However, in this case, the constant must be less than the distance between the two fixed points. A hyperbola is the set of points in a plane such that the difference of the distances from the two fixed points to any point in the set stays constant. As shown, the points for which PF-PG = c lie on one side of the figure while the points for which PG-PF = c lie on the other side.

Let us find an equation of the hyperbola having foci F(-5,0) and G(5,0) and difference of focal radii 6. By the definition, we have

|PF - PG| = 6

Using the distance formula, we obtain

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Simplifying the expression, we have

x2 / 32 - y2 / 42 = 1

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