**P
a r a m e t r i c E q u a t i o n**

**By**

**Jane
Yun**

Let
*x* = *f* (*t*) and *y* = *g*
(*t*), where *f* and *g* are two functions
whose common domain is some interval I.
The collection of points defined by

(*x*, *y*)
= (*f* (*t*), *g* (*t*))

is called a plane
curve. The equations

*x* = *f*
(*t*) *y* = *g* (*t*)

where *t*
is in *I*, are called parametric equations of the curve. The variable t is called a parameter.

Parametric
equations are particularly useful in describing movement along a curve. Suppose that a curve is defined by the
parametric equations

*x* = *f *(*t*), *y* = *g *(*t*), *a* ² *t* ² *b*

where *f*
and *g* are each defined over the
interval *a* ² *t* ² *b*. For a given value of *t*, we can find the value of *x* = *f*
(*t*) and *y* = *g* (*t*), obtaining point (*x*, *y*)
on the curve. In fact, as *t* varies
over the interval from *t* = *a* to *t*
= *b*, successive values of *t* give rise to a directed movement along
the curve; that is, the curve is traced out in a certain direction by the
corresponding succession of points (*x*,
*y*).

Now, letÕs investigate parametric equation

*x*** = a cos (t)**

*y*** = b sin (t)**

by varying *a* and *b*.

LetÕs begin exploration when ** a
= b**.

Notice that the curve is a circle with
center at (0, 0) and radius *a*. As the parameter *t* increases, say from *t* = 0[the point (*a*, 0)] to *t* = Π/2
[the point (0, *a*)] to *t* = Π [the point (-*a*, 0) to *t* = 3Π/2 [the point (-*a*,
0)], we see that the corresponding points are traces in a counterclockwise
direction around the circle. Since
a = b, as 'a', which is the radius, changes, it just changes the radius of a circle.

The presence of sines and cosines in the
parametric equations suggests that we use a Pythagorean identity. In fact, since

**cos t = x/a sin t = y/a**

we find that

**cos ^{2} t +
sin^{2} t = 1**

**( x/a)^{2} + (y/a)^{2} = 1**

*x*^{2}** + y^{2} = a^{2}**

^{ }

LetÕs discuss the curve further. The domain of each parametric equations
is -° < *t* < °. Thus, the
graph is actually being repeated each time that t increases by 2п.

If we wanted the curve to consist of
exactly 1 revolution in the counter clockwise direction, we could write

*x*** = a cos t, y = a sin t, 0 ² t ²
2п**

This curve stats at *t* = 0 [the point (*a*, 0)]
and, proceeding counterclockwise around the circle, ends at *t* = 2п [also the point (*a*, 0)].

Next, letÕs try when ** a > b**.

Let *a* = 2 and *b* = 1.*
*

The curve is an ellipse with center (0, 0)
and foci at (-*c*,0) and (*c*, 0). The foci are labeled F_{1} and F_{2}. The line containing the foci is called
the major axis. The midpoint of
the line segment joining the foci is the center of the ellipse. The line through the center and
perpendicular to the major axis is the minor axis.

The two points of intersection of the
ellipse and the major axis are the vertices, V_{1} and V_{2},
of the ellipse. The distance from
one vertex to the other is the length of the major axis. The ellipse is symmetric with respect
to its major axis, with respect to the minor axis, and with respect to the
center.

Again, since The presence of sines and
cosines in the parametric equations suggests that we use a Pythagorean
identity. In fact, since

**cos t = x/a sin t = y/b**

we find that

**cos ^{2} t +
sin^{2} t = 1**

**( x/a)^{2} + (y/b)^{2} = 1 **

where *a*
> *b* > 0 and *b*^{2} = *a*^{2} – *c*^{2}. This is the ellipse equation with
center at (0, 0) and foci at (-c, 0) and (+*c*,
0).

As you can verify, the ellipse defined by
equation above is symmetric with respect to the x-axis, y-axis, and origin.

Because the major axis is in the *x*-axis, we find the vertices of the
ellipse defined by equation above by letting *y* = 0. The vertices
satisfy the equation **( x/a)^{2}
= 1**, the solution of which are x = +

For further investigation, letÕs try when **a < b**.

IÕm going to let a = 1 & b = 2.

This is figure illustrates the graph of
such an ellipse. Notice the right
triangle with the points at (0, 0), (b, 0), and (0, c). Now, letÕs find the
equation of an ellipse when the major axis is in the *y*-axis in terms of *x* and *y*.

Since

**cos t = x/b sin t
= y/a**

we find that

**cos ^{2} t +
sin^{2} t = 1**

**( x/b)^{2} + (y/a)^{2} = 1**

where *a*
> *b* > 0 and *b*^{2} = *a*^{2} – *c*^{2}.
An equation of this ellipse is
center at (0, 0) and foci at (0, -*c*)
and (0, +*c*). The major axis is the *y*-axis; the vertices are at (0, -*a*) and (0, *a*).

** **

Look closely at equations when *a* > *b* and when *a* < *b*.
Although they may look alike, there is a difference. In equation when *a* > *b*, the larger
number, *a*^{2}, is in the
denominator of the *x*^{2}-term,
so the major axis of the ellipse is along the *x*-axis. In equation
when *a* < *b*, the larger number, a^{2}, is the in the denominator of
the *y*^{2}-term, so the major
axis is along the *y*-axis.