The Hyperbola

• Definition and equation
  
• Orthogonal hyperbola as a special hyperbola
  
• Parametric equations of the hyperbola
  
• Geometric property of a point of the hyperbola
  
• Tangent line in a point P of a hyperbola
  
• Properties
  
• Solved problems
  

Definition and equation

 
        x2   y2
        -- - --- = 1
        a2   b2

 
        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

 

    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

 
                x2  - y2  = a2

Parametric equations of the hyperbola

 
        x = a sec(t)            (1)
        y = b tan(t)            (2)

 
              a
        x = ------
            cos(t)

            b sin(t)
        y = -------
             cos(t)

 
                 a
        cos(t) = -
                 x

                 a y
        sin(t) = ---
                 b x

 

        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

 
        P(a sec(t) , b tan(t))

Geometric property of a point of the hyperbola

 
  |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

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

 
         xo x   yo y
         ---- - ---- = 1
          a2     b2

Properties

• The y-intercepts of the rectangle are (0,b) and (0,-b).
• The x-intercepts of the circle are the foci.

• 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.

Solved problems

• 
  

  A hyperbola H has vertices P and P' and D is on H. Prove that the product of the slopes of DP and DP' is constant.

  
  

  Take D(a sec(t) , b tan(t)) on H.
  The slope of DP is

   
            b tan(t)
          -------------
          a(sec(t)- 1)
  

  The slope of DP'is

   
            b tan(t)
          -------------
          a(sec(t)+ 1)
  

  The product of the slopes of DP and DP' is

   
            b2tan2(t)         b2
          ---------------  = -----
          a2(sec2(t)- 1)      a2
  

• 
  

  Calculate the lines with slope = 1 and tangent to the hyperbola 9 x2 - 25 y2 = 225 Find the equation of the line connecting the points of tangency.

  
  

  Asw: y = x + 4 ; y = x - 4 ; 9x - 25 y = 0

• 
  

  A point P(xo,yo) is on the hyperbola H with foci F and F'. Prove that |D,F| = | a - (c/a) xo |.

  
  

  Take x_o = a sec(t) and y_o = b tan(t) . Then P(x_o,y_o) is on the hyperbola H .

   
          |D,F|2=  (c - a sec(t))2 + b2 tan2(t)
  
                =  (c - a sec(t))2 + (c2 - a2) (sec2(t) - 1)
  
                = ...
  
                = a2 - 2 a c sec(t) + c2 sec2(t)
  
                = (a - c sec(t))2
  
                        c xo
                =  (a - ----)2
                         a
  

• 
  

  Find the equations of the tangent lines, with slope m, to a hyperbola.

  
  

  The equation of a line with slope m is y = m x + q. It suffices to determine q so that the line and the hyperbola have two coincident intersection points.

  The intersection points are the solutions of the system

   
          y = m x + q
  
          x2   y2
          -- - --- = 1
          a2   b2
  

  We substitute y from the first equation in the second equation

   
      x2       (m x + q)2
     ----   - -------------- = 1
      a2         b2
  

  We transform the equation to the standard form of a quadratic equation. After simplification it is

   
    (b2 - m2 a2) x2 - 2 a2 q m x - b2 a2 - q2 a2 = 0
  

  The line y = m x + q is tangent line if and only if the discriminant of the quadratic equation is zero. After simplification the condition is :

   
        a2 b2 q2 + a2 b4 - a4 b2 m2 = 0
  <=>
        q2  = a2 m2 - b2
  

  The tangent lines are

   
        y = mx + sqrt( a2 m2 - b2 )
  and
        y = mx - sqrt( a2 m2 - b2 )
  
  
  

• 
  

  Point P is on the hyperbola x2/a2 - y2/b2 = 1. Take on an asymptote s point Q with the same abscissa as point P. The line l is the perpendicular on s through point Q and the line n is the normal line in point P. Show that l and n intersection on the x-axis.

  
  

  Take P(a/cos(t) , b tan(t)) and Q(a/cos(t) , b/cos(t)).
  Line l has an equation

   
  y - b/cos(t) = (-a/b) (x - a/cos(t))       (*)
  

  The tangent line in P is (1/(a cos(t)) - y (tan(t)/b) = 1.
  After simplification the slope of the tangent line is b/(a sin(t)).
  The slope of n is (-a/b)sin(t).
  The equation of the normal n in P is

   
     y - b.tan(t) = (-a/b)sin(t) (x - a/cos(t)).
  <=>
     y - b sin(t)/cos(t) = (-a/b) sin(t) (x - a/cos(t))
  <=>
     y/sin(t)- b/cos(t) = (-a/b) (x - a/cos(t))    (**)
  

  Now, compare the equations (*) and (**).
  Without any calculation we see that, for y = 0, the x-value is the same.

• 
  

  Find the m-values such that the angle between the asymptotes of the hyperbola x2/(m+1)2 - y2/m2 = 1 is equal to 60 degrees.

  
  

  a = m+1 en b = m.

  • The angle between the asymptote y = bx/a and the x-axis is 30 degrees.

    Then the slope b/a = 1/sqrt(3) <=> m/(m+1) = 1/sqrt(3) <=> ... <=> m = sqrt(3)/(3-sqrt(3))

  • The angle between the asymptote y = bx/a and the x-axis is 60 degrees.

    Then the slope b/a = sqrt(3) <=> m/(m+1) = sqrt(3) <=> ... <=> m = sqrt(3)/(1-sqrt(3))

• 
  

  Point P is on the orthogonal hyperbola x2 - y2 = a2. Point P' is the perpendicular projection of P on the x-axis. Show that |PP'|2 is equal to the power of point P' relative to a circle x2+y2=a2.

  
  

  P(a sec(t) , a tan(t)) and P'(a sec(t) , 0)
  |PP'| = a² tan²(t)
  The power of point P' relative to a circle x²+y²=a² is :

   
      (a sec(t) - 0)2 + (0-0)2 -a2   (see theory circle)
  
      = a2 sec2(t) - a2 = a2( sec2(t) - 1) = a2  tan2(t)
  
  

• 
  

  Prove that the slopes m of the lines through P(xo,yo) and tangent to the hyperbola b2 x2 - a2 y2 = a2 b2 are the solutions of the equation (xo2 - a2) m2 - 2 xo yo m + yo2 + b2 = 0

  
  
• 
  

  The tangent line in point P of a hyperbola has with the asymptotes the intersection points Q and Q'. Show that P is the midpoint of the segment [Q,Q'].