Understanding Conic Sections: Ellipse, Parabola, and Hyperbola Explained

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Homework Help Overview

The discussion centers around the classification of conic sections based on the general equation Ax^2 + Cy^2 + Dx + Ey + F = 0. Participants are exploring how the conditions AC > 0, AC = 0, and AC < 0 correspond to ellipses, parabolas, and hyperbolas, respectively, while also considering degenerate cases.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss completing the square and rearranging the equation to analyze different cases. There are questions about how to handle cases where A or C are zero and the implications of these conditions on the resulting curves.

Discussion Status

Some participants have suggested starting with completing the square for non-zero coefficients, while others are attempting to clarify the process and share their progress. There is an ongoing exploration of the implications of different values for A and C, and the discussion is actively developing without a clear consensus yet.

Contextual Notes

Participants note the challenge of handling degenerate cases and express uncertainty about the initial steps required to approach the problem effectively. There is also mention of difficulties with formatting mathematical expressions, which may affect clarity in communication.

PotatoSalad
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Here's the question:
Consider the equation:
Ax^2+Cy^2+Dx+Ey+F=0

Consider the cases AC>0, AC=0 and AC<0 and show that they lead to an ellipse, parabola and hyperbola respectively, except for certain degenerate cases. Discuss these degenerate cases and the curves that arise from them

Don't really know where to start. I can 'prove' it by doing examples but that is not sufficient. Can somebody get me started on answering this question please?
Thanks
 
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Complete the squares, and rearrange in the cases where neither A or C are zero.
Then tackle the cases where either A, C or both are zero.
 
Ok... so I complete the square to:
EDIT: I suck at using LaTeX. Give me a minute...

Ok here it is:
http://img296.imageshack.us/img296/4413/eqnyo7.jpg
 
Last edited by a moderator:
In the case of non-zero A and C, it is simpler to do it the following way, in order to not get into the silly trouble of square-rooting negative numbers:
A(x+G)^{2}+C(y+H)^{2}=I, G=\frac{D}{2A}, H=\frac{E}{2C}, I=AG^{2}+CH^{2}-F
 
Last edited:

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