Themes > Science > Mathematics > Algebra > Foci of a conic section > Topics and Problems > Analytic > The Hyperbola

..Definition and equation
..Orthogonal hyperbola as a special hyperbola
..Parametric equations of the hyperbola
..The hyperbola and to functions
..Tangent line in a point D of a hyperbola
..Properties


Definition and equation

Take two points F and F' and a strictly positive value 2a.
The locus of all points D such that abs(|D,F| - |D,F'|) = 2a is a hyperbola.
Here abs(x) means absolute value of x.
We choose the line FF' as x-axis and the perpendicular bisector of the segment [F,F'] as y-axis.
We give F and F' resp. coordinates (c,0) and (-c,0).

We see that a
The points F and F' are called the foci of the hyperbola. 

 
        D(x,y) is on the hyperbola
                <=>
        |D,F| - |D,F'| =  (+ or -) 2a
                <=>
  _______________      _______________
 |        2    2      |        2    2
\| (x - c)  + y   -  \| (x + c)  + y    =  (+ or -) 2a

Squaring        <=>
                                    ________________________________
       2    2          2    2      |         2    2          2    2      2
(x - c)  + y  + (x + c)  + y  - 2 \| ((x + c)  + y ) ((x - c)  + y ) = 4a

                <=>
                ...

                <=>
  ________________________________
 |         2    2          2    2
\| ((x + c)  + y ) ((x - c)  + y ) = x2  + y2  + c2  - 2 a2

                <=>
  _______________________________________________
 |   2    2    2            2    2    2
\| (x  + c  + y  + 2 c x) (x  + c  + y  - 2 c x) =(x2  + y2  + c2 )- 2 a2    (*)

Squaring        =>


(x2 + y2  + c2 )2 - 4 x2  c = 4 a4  - 4 a2  (x2 + y2  + c2 ) + (x2  + y2  + c2 )2    (**)

                <=>

        - 4 x2  c = 4 a4  - 4 a2  (x2  + y2  + c2 )
                <=>
                ...

                <=>

        (c2  - a2 ) x2  - a2  y2  = a2  (c2  - a2 )


                       Since a < c , we can say  c2  - a2  = b2

                <=>

        b2  x2  - a2  y2  = a2  b2

                <=>
          x2    y2
          -- - -- = 1           (***)
          a2    b2
This is the equation of the hyperbola.
But we don't know if (*) and (**) are equivalent because we don't know if the right side of (*) is a positive value.
From above we know that if |D,F| - |D,F'| = 2a then (***) holds.
To prove the reverse, it is sufficient to show that if (***) holds, then the right side of (*) is a positive value.
Well, from (***) we have 
 
        x2
        -- > or = 1    =>   x2 > or = a2
        a2

                but also    c2  >  a2
                        -------------------
                        x2  + c2  > 2 a2


        => x2  + y2  + c2  > 2 a2

        then the right side of (*) is a positive value.
Now abs(|D,F| - |D,F'|) = 2a and (***) are equivalent.
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 [D,F'] and [D,F] are the focal radii through point D.

Orthogonal hyperbola as a special hyperbola

If a = b the equation becomes 
 
                x2  - y2  = a
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 
 
        D(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.

The hyperbola and to 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
We see that the hyperbola defines two functions.
The graph of the first one is above the x-as and the second graph is the reflection of the first graph with respect to the x-axis. If we calculate the asymptotes a and a', we find 
 

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

Tangent line in a point D of a hyperbola

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

Properties

It is easy to prove, with previous figure, that:
  • The y-intercepts of the rectangle are (0,b) and (0,-b).
  • The x-intercepts of the circle are the foci.
In the same way as for the ellipse, it is easy to prove that:
  • The mirror image of F with respect to a variable tangent line is on the circle with center F' and with radius 2a.
  • The orthogonal projection of F on a variable tangent line is the circle with center O and radius a.


Information Provided by Johan.Claeys@ping.be