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### Uses for Parametric Equations

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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.
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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

So:

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
Check out our web site! http://mathforum.org/dr.math/
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Associated Topics:
High School Equations, Graphs, Translations

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