Parametric Equations
      Parametric equations of a hyperbola
         The parametric equations of a hyperbola expressed by trigonometric functions
         The parametric equations of a hyperbola expressed by hyperbolic functions
         The definition of the hyperbolic functions
         Relations between hyperbolic and trigonometric or circular functions
         The parametric equations of the equilateral or rectangular hyperbola
         The parametric equations of the general hyperbola
Parametric equations of a hyperbola
In the construction of the hyperbola, shown in the figure below, circles of radii a and b are intersected by an arbitrary line through the origin at points M and N.
Tangents to the circles at M and N intersect the x-axis at R and S.
On the perpendicular to the x-axis through S mark the line segment SP of the length MR to get the point P
of the hyperbola.
Let prove that P is a point of the hyperbola.
In the right triangles ONS and OMR,
by replacing OS = x and  MR = SP y 
and substituting
by dividing by b2,
therefore, P(x, y) is the point of the hyperbola.
The coordinates of the point P(x, y) can also be expressed by the angle t common to both triangles ONS and OMR, therefore
 
x = a sec t
 y = b tan t
 
are the parametric equations of the hyperbola expressed by trigonometric functions.
By substituting these parametrically expressed coordinates into equation of the hyperbola
that is, known trigonometric identity.
The parametric equations of a hyperbola expressed by hyperbolic functions
The definition of the hyperbolic functions
The hyperbolic functions are defined in terms of exponential functions ex and  e-x as
  -hyperbolic sine,   -hyperbolic cosine,
 -hyperbolic tangent,    -hyperbolic cotangent,
 -hyperbolic secant and  - hyperbolic cosecant.
By adding and subtracting formulas for the sinh x and the cosh x, we get 
cosh x + sinh x = ex
cosh x - sinh x = e-x,
then multiplying the two equations obtained is
  cosh2 x - sinh2 x = 1  
the basic identity of the hyperbolic functions.
Relations between hyperbolic and trigonometric or circular functions
Let compare the basic identity of the hyperbolic functions with the trigonometric identity obtained from the parametric equations of the hyperbola expressed by trigonometric functions shown above,
cosh2 x - sinh2 x = 1
sec2 a - tan2 a = 1
therefore, we can write
cosh x = sec a   and    sinh x = tan a
by substituting into
The parametric equations of the equilateral or rectangular hyperbola
The parametric equations
 
x = cosh t
y = sinh t
 
describe the right branch of the rectangular hyperbola,
with semi-axes a = b = 1 centered at the origin, while
the parameter t passes through all real values from
 - oo  to + oo .
Thus, by squaring both equations then, subtracting the
second equation from the first and applying the basic
identity, obtained is
x2 - y2 = 1  the rectangular hyperbola.
The above figure shows the construction of a point P(x, y) of the rectangular hyperbola using the relations between hyperbolic and trigonometric functions.
Use the hypotenuse of the right triangle O1P' and the side opposite to the angle a to get the abscissa and the ordinate of a point of the hyperbola respectively, as
x = sec a = cosh t   and   y = tan a = sinh t.
By changing values of the angle a from the interval  - p/2 < a < p/2  we can get every point of the hyperbola.
The parametric equations of the general hyperbola
The parametric equations
 
x = a cosh t
 y = b sinh t,
     - oo < t < + oo
represent the general hyperbola since plugging the coordinates x and y into the equation of the hyperbola
yield the basic identity  cosh2 x - sinh2 x = 1.
Pre-calculus contents C
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