Parametric Equations

Lines

Recall that a line has equation

y  =  mx + b

Suppose that one airplane moves along the line

y  =  2x + 3

while the other airplane moves along the line

y  =  3x - 2

Can you tell whether the airplanes collide?  Even though the lines intersect, the equations themselves do not tell us whether there will be a mid air collision.  To be able to mathematically model this scenario, we use parametric equation.  We introduce the variable t for time and write x and y as a function of t.

Consider the two sets of equations:

1. x(t) = t,       y(t) = 2t + 1

2. x(t) = 2t,     y(t) = 4t + 1

These describe the same line, but the second one travels twice as fast.

 Definition  A curve given by            x = x(t), y = y(t)  is called a parametrically defined curve and the functions            x = x(t) and y = y(t)  are called the parametric equations for the curve.

Finding the Parametric Equations for a Line Given Two Points

Example:

Find the parametric equations for the line through the points (3,2) and (4,6) so that when t = 0 we are at the point (3,2) and when t = 1 we are at the point (4,6).

Solution:

We write symbolically:

(x,y)  =  (1 - t) (3, 2)+ (t) (4, 6)

=  (3 - 3t + 4t, 2 - 2t + 6t)  =  (3 + t, 2 + 4t)

so that

x(t)  =  3 + t     and     y(t)  =  2 + 4t

If y = f(x) is a function of x we can write parametric equations by writing

x  =  t      and      y  =  f(t).

Example

The parabola

y = x2

can be represented by the parametric equations:

x  =  t     and     y  =  t2

Consider the circle centered at (0,0) with radius 2.  We can write it parametrically as

x(t)  =  2cos(t)     and     y  =  2sin(t)

We see that the circle is drawn in a counterclockwise direction.  We can draw the same circle as

x(t)  =  2cos(-t)     and     y(t)  =  2sin(-t)

now the circle is drawn clockwise.  We can also write

x(t)  =  2 cos(t2)     and     y  =  2 sin(t2)

now the circle begins slowly and speeds up.

A Cool Example

The graph of

x(t) = 11cost - 6cos(11/6 t)      and      y(t) = 11sin(t) - 6sin(11/6 t)

is pictured below:

Eliminating the Parameter

If a curve is given by parametric equations, we often are interested in finding an equation for the curve in standard form:

y = f(x)

Example

Consider the parametric equations

x(t) = t2     and     y(t) = sin(t)      for  t  >  0

To find the conventional form of the equation we solve for t:

t =

hence

y = sin()

is the equation.

Example

Eliminate the parameter for

x(t)  =  et     and     y(t)  =  e2t + 1

Solution

We write:

y(t)  =  (et)2 + 1

Hence

y = x2 + 1

Let

x1(t)  =  2t + 1     and     y1(t)  =  4t2

and

x2(t)  =  3t      and      y2(t)  =  3t

Do they intersect?  If so then there is a c with

2c + 1  =  3c

and

4c2  =  3c

the first equation gives us that

c  =  1

Putting this into the second equation we have

4  =  3

which tells us that they do not intersect.  Do their graphs intersect?  If so then there exists a c and a k such that

2c + 1  =  3k

and

4c2   =  3k

Hence, we see that

2c + 1  =  4c2

or that

4c2  - 2c - 1  =  0

We solve to get two intersection points

hence their graphs intercept.  Their graphs are shown on the right.

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