Hyperbolas

    In general, a hyperbola is a type of conic section defined by the intersection of a right circular conical surface and a plane which cuts through both halves of the cone. In this topic we define a hyperbola as locus of points where the difference from any point on the curve to two fixed points (the foci) is a positive constant. We illustrate the definition with several examples including how to find an equation of a hyperbola given some geometric information. Conversely, we also show how to find the vertices, major axis, minor axis, eccentricity, and focus given the equation of the hyperbola.  

Definition (Hyperbola)  A hyperbola is the set of points in a plane, the difference of whose distances from two fixed points (the foci) in the plane is a positive constant.

Proposition (Hyperbola)  

    (i) The graph of is a hyperbola centered at the origin and has foci where The length of the transverse axis is the length of the conjugate axis is and the asymptotes are
    
    (ii) The graph of is a hyperbola centered at the origin and has foci where    The length of the transverse axis is the length of the conjugate axis is and the asymptotes are

Proposition (Hyperbola)  

    (ii) The graph of is a hyperbola centered at and has foci where The length of the transverse axis is the length of the conjugate axis is and the asymptotes are
    
    (ii) The graph of is a hyperbola centered at and has foci where    The length of the transverse axis is the length of the conjugate axis is and the asymptotes are

Example (Hyperbola)  Find the vertices, foci, and asymptotes of the hyperbola

    Solution. We have the form and with and so and Thus, Therefore, the vertices are and the foci are and the asymptotes are The hyperbola is centered at the origin.


    
    
    
Example (Hyperbola)  Find the vertices, foci, and asymptotes of the hyperbola  

    Solution.  The hyperbola is centered at and we have the form and with and so and Thus, Therefore, the vertices are and the foci are and the asymptotes are


    
       
    
Example (Hyperbola)  Find the vertices, foci, and asymptotes of the hyperbola  

    Solution. First we need to complete the square in and We have,















Therefore, we have So we have the form with , and Thuis, and so Our vertices are at , the foci are and the asymptotes are



    

Example (Find the Equation of the Hyperbola)  Find an equation of the hyperbola that is centered at the origin and satisfies the geometric conditions: the vertices are and the hyperbola passes through the point

    Solution. Since the vertices are we know that the hyperbola has a horizontal axis and is centered at the origin. Also, an equation has the form where To find we use the point in the form We find that and so an equation is
         
Example (Find the Equation of the Hyperbola)  Find an equation of the hyperbola that is centered at the origin and satisfies the geometric conditions: the hyperbola has a horinzontal transverse axis of length 6 and conjugate axis of length 2.

    Solution. Since we have a horizontal axis we will use the form Since the length of the transverse axis is we have The length of the conjugate axis being 14 means that and so we have
    
Example (Find the Equation of the Hyperbola)  Find an equation of the hyperbola that is centered at the origin and satisfies the geometric conditions: the foci of the hyperbola are and the difference of the distances from a point on the hyperbola to the foci is 8.

    Solution. Since the foci are we know that the hyperbola has a horizontal axis, is centered at the origin, and an equation has the form where The difference of the distances from a point on the hyperbola to the foci is 8. Si, in particular if we consider a vertices we have and so Thus and we have   

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Hyperbolas
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/hyperbolas.html