The Hyperbola




Definition and equation

In an appropriate coordinate system the hyperbola is a curve with an equation of the form
 
        x2   y2
        -- - --- = 1
        a2   b2
In this equation a and b are strictly positive real numbers. The equation consists of two functions.
 
        x2   y2
        -- - --- = 1
        a2   b2

<=>
               b2  (x2  - a2 )
        y2  = -----------------
                    a2
<=>
                 _________                   _________
           b    |  2    2               b   |  2    2
       y = -   \| x  - a     or   y = - -  \| x  - a
           a                            a
The graph of the first one is above the x-as and the second graph is the reflection of the first graph in the x-axis. If we calculate the asymptotes a and a', we find
 

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

The intersection points of the hyperbola with the x-axis are A'(-a,0) and A(a,0). These are the vertices on the x-axis.
The segment [A',A] is called major axis of the ellipse.
The segments [P,F'] and [P,F] are the focal radii through point P.

Orthogonal hyperbola as a special hyperbola

If a = b the equation becomes
 
                x2  - y2  = a2
This hyperbola is called an orthogonal hyperbola.

Parametric equations of the hyperbola

Take in a plane two lines l and m with resp. equations
 
        x = a sec(t)            (1)
        y = b tan(t)            (2)
The real number t is the parameter.
We know, from the theory of 'Elimination of parameters', that the intersection points of the two associated lines constitute a curve. To obtain the equation of that curve, we eliminate the parameter t from the two equations. This means that we search for the condition such that (1) and (2) has a solution for t.
The simultaneous equations (1) and (2) are equivalent to
 
              a
        x = ------
            cos(t)

            b sin(t)
        y = -------
             cos(t)
or to
 
                 a
        cos(t) = -
                 x

                 a y
        sin(t) = ---
                 b x
This system has a solution for t if and only if
 

        sin2 (t) + cos2 (t) = 1
<=>
         a 2    a y 2
        (-)  + (---)  = 1
         x      b x
<=>

        a2  b2  + a2  y2  = b2  x2
<=>
        x2   y2
        -- - --- = 1
        a2   b2
Hence, the two associated lines constitute a curve and that curve is the hyperbola.
We say that (1) and (2) are parametric equations of the hyperbola.
The point
 
        P(a sec(t) , b tan(t))
is on the hyperbola for each t-value and with each point of the hyperbola corresponds a t-value.

Geometric property of a point of the hyperbola

We start with P(a sec(t) , b tan(t)).
 
  |PF|2 = (x-c)2 + y2

               a                 b2 sin2t
         = ( -----  - c )2  + -----------
             cos t               cos2t

              1
         = --------- ( a2 - 2 a c cos t + c2 cos2 t + (c2-a2) sin2t )
            cos2 t

             1
        = --------- ( a2 cos2t  - 2 a c cos t + c2)
           cos2 t

             1
        = --------- ( a cos t - c)2
           cos2 t


   |PF|2 = (x+c)2 + y2

          = ....

                1
          = --------- ( a cos t + c)2
            cos2 t

    Since  c > a  we have

    |PF| = (1/cos(t)).(c - a cos(t))

    |PF'| = (1/cos(t)).(c + a cos(t))

    |PF'| - |PF| = 2a
In the same way as for the ellipse you can show that if the difference of the distances from P to F and F' is equal to 2a, the point P is on the hyperbola. (exercise)

The hyperbola is the locus of the point P such that the difference of the distances from P to F and F' is equal to 2a

Tangent line in a point P of a hyperbola

In the same way as for the ellipse you'll find the equation of the tangent line in point P(xo,yo). (exercise)
 
         xo x   yo y
         ---- - ---- = 1
          a2     b2
It is the bisector t of the lines PF and PF'.

Properties

With the help of the previous figure you can show that: In the same way as for the ellipse you can , as an exercise, prove that:

Solved problems




Topics and Problems

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