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Uses for Parametric Equations

Date: 04/23/98 at 21:12:07
From: Brian Immerman
Subject: Parametric equations

What is a parametric equation used for? I have had a lot of trouble 
trying to find the answer to this somewhat simple question.

Date: 04/24/98 at 07:38:03
From: Doctor Anthony
Subject: Re: Parametric equations

A simple example is the equation of a parabola: y^2 = 4ax.

If we use the parametric form  x=at^2, y = 2at, then I have reduced 
the number of variables from 2 (x, y) to 1 (t).

Any value of t now defines a unique point on the parabola, and this 
parameter 't' is used when, for example, finding where other lines or 
curves cut the parabola. The slope of the parabola is 2a/(2at) = 1/t  
at any point t, and the equation of the tangent is:

          y-2at = 1/t (x-at^2)

     ty - 2at^2 = x - at^2

             ty = x + at^2

Thus for a value of t, we have the equation of the tangent at that 
point on the parabola. The simplification of algebraic work when 
working in coordinate geometry is considerable if curves are expressed 
in parametric form.

A final example concerns normals to a parabola. Show that from any 
point (p,q) three normals can be drawn to a parabola.

The equation of a normal will be:

                  y-2at = -t(x-at^2)

                  y-2at = -tx + at^3


     at^3 + t(2a-x) - y = 0     

And if a normal passes through (p,q):

     at^3 + t(2a-p) - q = 0

This is a cubic in t, so in general there are 3 points t1, t2, t3 
where a normal from (p,q) can be drawn. Of course, a cubic might have 
one real and two complex roots, so there are points (p,q) from which 
only ONE real normal could be drawn.

The point to note, however, is how easy this question was because we 
used the parametric form for the parabola. As an exercise you might 
try proving the result without the use of a parameter.

-Doctor Anthony,  The Math Forum
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Associated Topics:
High School Equations, Graphs, Translations

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