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An Ellipse Or A Circle? - Parametric Equations

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Date: 12/05/98 at 18:49:16
From: Jacob
Subject: Parametric equations

Hi Doctor Math,

My friend and I are in a debate over a parametric equation of:

x = 4(1-sin t), y = 4(1-cos t) over the interval 0 < t < 2pi.

I believe it is elliptical in the first coordinate and he says it is a
circle in the first coordinate with center (4,4). What does it look
like?

Also we want to figure the slopes at points 0, pi/4, pi/2, 3pi/2, and
2pi? If it is a circle the answers are obvious, but if it's not, what
would they be?

Jacob Sands
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Date: 12/05/98 at 19:30:09
From: Doctor Barrus
Subject: Re: Parametric equations

Hi, Jacob!

It's pretty easy to recognize conic sections by their equations when
the equations are written in terms of x and y. What you can try to do,
then, is come up with an equation in terms of x and y - eliminate all
t's. There are a number of ways we can do this. I'm going to use the
fact that

(sin(t))^2 + (cos(t))^2 = 1

Solving your equations for sin(t) and cos(t), we get

x = 4(1-sin t)   =>   sin t = -(x-4)/4
y = 4(1-cos t)   =>   cos t = -(y-4)/4

Then (sin t)^2 + (cos t)^2 = 1 means that

[-(x-4)/4]^2 + [-(y-4)/4]^2 = 1

(x-4)^2   (y-4)^2
------- + ------- = 1
16        16

(x-4)^2 + (y-4)^2 = 16, or
x^2 + y^2 - 8x - 8y + 16 = 0

Since t ranges from 0 to 2pi, we're going to cover all possible sines
and cosines, so we don't really have to worry about t's interval. From
either of the last two equations above you can recognize that the conic
section is a circle. The top one fits the form

(x-h)^2 + (y-k)^2 = r^2

and so tells you that the circle is centered at (h,k) = (4,4) and has
a radius of r = 4. This would place the circle in the first quadrant.

It sounds like you know how to find the derivative at the given points,
so I'll let you figure that out. If it turned out to be an ellipse,
though, what you could do is use implicit differentiation on the x-y
equation you arrived at at the end to find an equation for dy/dx.
Knowing t, you could then use the equations you began with to figure
out the appropriate x and y to plug into the derivative equation.

I hope this has helped. If you have further questions, feel free to
write back. Good luck!

- Doctor Barrus, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Basic Algebra
High School Calculus
High School Conic Sections/Circles
High School Equations, Graphs, Translations
High School Geometry

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