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Conic Sections

Hyperbolas

Problems

Problems

A hyperbola is the set of all points such that the difference of the distances between any point on the hyperbola and two fixed points is constant. The two fixed points are called the foci of the hyperbola.

Figure %: The difference of the distances d 1 - d 2 is the same for any point on the hyperbola.

The graph of a hyperbola is not continuous--every hyperbola has two distinct branches. The line segment containing both foci of a hyperbola whose endpoints are both on the hyperbola is called the transverse axis. The endpoints of the transverse axis are called the vertices of the hyperbola. The point halfway between the foci (the midpoint of the transverse axis) is the center.

The standard equation for a hyperbola with a horizontal transverse axis is - = 1 . The center is at (h, k) . The distance between the vertices is 2a . The distance between the foci is 2c . c 2 = a 2 + b 2 . The line segment of length 2b perpendicular to the transverse axis whose midpoint is the center is the conjugate axis of the hyperbola.

The standard equation for a hyperbola with a vertical transverse axis is - = 1 . The center is at (h, k) . The distance between the vertices is 2a . The distance between the foci is 2c . c 2 = a 2 + b 2 .

Every hyperbola has two asymptotes. A hyperbola with a horizontal transverse axis and center at (h, k) has one asymptote with equation y = k + (x - h) and the other with equation y = k - (x - h) . A hyperbola with a vertical transverse axis and center at (h, k) has one asymptote with equation y = k + (x - h) and the other with equation y = k - (x - h) .

The eccentricity of a hyperbola, like an ellipse, is e = . For all hyperbolas, though, c > a , so e > 1 . If e is close to one, the branches of the hyperbola are very narrow, but if e is much greater than one, then the branches of the hyperbola are very flat.

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