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    Conic Sections: Hyperbola


    A hyperbola is defined as the locus of points where the difference in the distance to two fixed points (called foci) is constant. That fixed difference in distance is two times a where a is the distance from the center of the hyperbola to the vertex of the nearest branch of the hyperbola. a is also known as the semi-major axis of the hyperbola. The foci lie on the transverse axis and their midpoint is called the center.

    In an x-y coordinate system, the hyperbola opening east-west with center (h,k) along with semi-major axis, a, and semi-minor axis b is represented by:


    In an x-y coordinate system, the hyperbola opening north-south with center (h,k) along with semi-major axis, a, and semi-minor axis b is represented by:





    In the above figure, the blue colored hyperbola opens east-west while the light green colored hyperbola opens north-south.


    Eccentricity of Hyperbola:




    The foci for an east-west opening hyperbola are given by

    (h ± c, k) where c2 = a2 + b2

    and for a north-south opening hyperbola are given by

    (h, k ± c) where c2 = a2 + b2


    Hyperbolas In Polar Coordinates:

    East-west opening hyperbola:

    r2 = a sec 2θ

    North-south opening hyperbola:

    r2a sec 2θ

    Northeast-southwest opening hyperbola:

    r2 = a csc 2θ

    Northwest-southeast opening hyperbola:

    r2a csc 2θ

    In all formulas the center is at the pole, and a is the semi-major axis and semi-minor axis.


    Hyperbolas In Parametric Form:

    East-west opening hyperbola:

    x = a sec t + h,     y = b tan t + k

    or

    x = ± a cosh t + h,     y = b sinh t + k


    North-south opening hyperbola:

    x = a tan t + h,     y = b sec t + k

    or

    x = a sinh t + h,     y = ±b cosh t + k

    In all formulae (h,k) is the center of the hyperbola, a is the semi-major axis, and b is the semi-minor axis.