Algebra/Chapter 17/Hyperbolas

Algebra Hyperbola Ellipse Probability right|200px| (h,k) = (0,0) right|200px|thumb Hyperbolas can be thought of as the "opposite" of ellipses - instead of the sum of distances from two foci, the difference is used. This leads to two curves that point in opposite directions. The major axis is 2a units long, and the transverse axis is 2b units long. When a2 + b2 = c2, eccentricity is again c/a. The foci are at either side of the major axis, c units away from the center. Hyperbolas come in conjugate pairs: An "east-west" hyperbola is written as: while a "north-south" one is written as: The east-west hyperbola is conjugate to the north-south one, and vice versa. The conjugates can be written in a unified form: where either sign gives a hyperbola and the opposite sign its conjugate hyperbola. As the curve progresses, it gets closer and closer to two lines called asymptotes. The slopes of the asymptotes are If b = a, then the asymptotes are perpendicular and their slopes are as in the second diagram. Rectangular hyperbolas thumb|right|200px| Hyperbolas x y = a for a positive and negative An alternate view of rectangular hyperbolas has the coordinate axes as asymptotes. In this case, the conjugate hyperbolas are related by reflection in an axis.