The Hyperbola

  1. Definition of the Hyperbola (Geometric)

    The final conic that we will study is called the hyperbola.

                        Definition  
    Let P and Q be two fixed points, and c be a constant.  Then the set of points in the plane such that 

              |QR - PR| = c 

    is a hyperbola.


    Let C be the center of the hyperbola and V be a vertex.  We define the eccentricity of a hyperbola as 


    e = CP/VP

    Notice that this is the same definition as the definition for the eccentricity of the ellipse.  Notice also that the eccentricity of a hyperbola is always greater than 1.  We call the two pieces of the hyperbola the branches of the hyperbola.  The line segment that connects the vertices is called the transverse axis, while the other axis is called the conjugate axis.  


  2. The Equation of the Hyperbola

    The standard equation of the hyperbola with center at (h,k) is

           

     The Standard Equation of a Hyperbola

    (x - h)2/a2  -  (y - k)2/b2  = 1


    When the center is at the origin the equation becomes

    The Hyperbola With Center at the Origin

    x2/a2  -  y2/b2  = 1


    Solving for y we get:

            y2/b2  = x2/a2  - 1

    which reduces to 

            y2  = x2b2/a2  - b2  

    or

            y = +- ( x2b2/a2  - b2)1/2

    If x is large the b2 term goes away, hence we get the equation

            y = +- b/a x

            y = b/a x     and     y = -b/a x 

     are the asymptotes of the hyperbola.

     

    Example:  

    Sketch the hyperbola 

            (x - 2)2/4 - (y + 1)2/9  = 1


    Solution: 

    We see that the center is at (2,-1) , the vertices are at (0,-1) and (4,-1), and the asymptotes  have slope y = 3/2 and -3/2.  

     

  3. Identifying Conics:

    We can tell which type of conic an equation represents by the signs in front of the square term, the number of square terms and the coefficients in front of the square terms. 

     

    • If there is only one square term then the conic is a parabola

    • If the signs differ then the conic is a hyperbola

    • If the signs and the coefficients are the same, then the conic is a circle

    • If the signs are the same but the coefficients differ, then the conic is an ellipse.

 

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