How to Parameterize an Ellipse with Offsets?

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Homework Help Overview

The discussion revolves around parameterizing an ellipse defined by the equation x²/a² + y²/b² - 2x/a - 2y/b = 0. Participants are exploring methods to express the variables in a parameterized form.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to parameterize the ellipse but encounters difficulties in solving for y. Some participants suggest using the method of completing the square to rewrite the equation. Others question the choice of parameterization and whether any specific values for x should be considered.

Discussion Status

Participants are actively engaging with the problem, with some offering guidance on completing the square and others exploring the implications of different parameter choices. There is a sense of collaboration as they seek to clarify the shape described by the equation and its potential parameterization.

Contextual Notes

There is an emphasis on understanding the shape of the ellipse and how it can influence the parameterization approach. The original poster's attempts and the responses indicate a focus on exploring various parameterization strategies without reaching a definitive conclusion.

tysonk
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How do I parameterize the following?
x^{2}/a^{2} + y^{2}/b^{2} -2x/a -2y/b = 0

I tried letting x =t or some other parameters but found it difficult to solve for y.
 
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Try to write

\frac{x^2}{a^2} - \frac{2x}{a} =\left( \frac{x}{a} -c \right)^2 +d

for some c,d to be determined. Do the same thing with the y terms. This is called completing the square.
 
Thanks!
So letting t=x I was able to get a parameterized equation for y and x.
Is there any obvious thing I should let x be. Or can I just set it to whatever and solve y in terms of that?
 
tysonk said:
Thanks!
So letting t=x I was able to get a parameterized equation for y and x.
Is there any obvious thing I should let x be. Or can I just set it to whatever and solve y in terms of that?

Do you know what shape is described by your equation? When you figure it out, you might find a nice parametrization in terms of trig functions.
 

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