

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 xaxis at
R
and S.

On the perpendicular to the xaxis
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 b^{2}, 


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 

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
e^{x}
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 = e^{x} 
cosh
x 
sinh x = e^{}^{x}, 
then
multiplying the two equations obtained is 

cosh^{2}
x 
sinh^{2 }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, 
cosh^{2}
x 
sinh^{2 }x = 1 
sec^{2}
a

tan^{2 }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 

describe the right branch of the
rectangular hyperbola, 
with semiaxes 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 
x^{2}

y^{2}
= 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 cosh^{2}
x 
sinh^{2 }x = 1. 








Precalculus
contents C 



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