Questions about conic sections

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To find the focii and directrices of a conic section, it is recommended to convert the equation to its standard form, as no simpler method is identified. For determining the type of conic, the discriminant B² - 4AC from the general form Ax² + Bxy + Cx² + Dx + Ey + F = 0 can be used: a negative value indicates a circle or ellipse, zero indicates a parabola, and a positive value indicates a hyperbola. The discussion also touches on the need for methods to handle degenerate conics. Overall, the focus is on seeking straightforward algorithms for dealing with conic sections.
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Given the equation of a conic section, how can I:

1) find its focii

2) find the equations of its directrices

3) find out what type of conic it is, without using either the arduous matrix method or the equally arduous rotation method

To be honest, I don't really like conic sections and I'm just looking for an algorithm for these.

Thanks for any help.
 
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gangsta316 said:
Given the equation of a conic section, how can I:

1) find its focii

2) find the equations of its directrices

3) find out what type of conic it is, without using either the arduous matrix method or the equally arduous rotation method

To be honest, I don't really like conic sections and I'm just looking for an algorithm for these.

Thanks for any help.
For both (1) and (2), I don't know of any method simpler than finding its standard equation. For 3, if the conic section is given in the general for Ax^2+ Bxy+ Cx^2+ Dx+ Ey+ F= 0, look at its "discriminant" B^2- 4AC. If it is negative, the conic section is either a circle or an ellipse. If it is 0, the conic section is a parabola. If it is positive, the conic section is a hyperbola.

(This doesn't really have anything to do with "Linear and Abstract Algebra" so I am moving it to "General Math".)
 
HallsofIvy said:
For both (1) and (2), I don't know of any method simpler than finding its standard equation. For 3, if the conic section is given in the general for Ax^2+ Bxy+ Cx^2+ Dx+ Ey+ F= 0, look at its "discriminant" B^2- 4AC. If it is negative, the conic section is either a circle or an ellipse. If it is 0, the conic section is a parabola. If it is positive, the conic section is a hyperbola.

(This doesn't really have anything to do with "Linear and Abstract Algebra" so I am moving it to "General Math".)

Thank you. For 3, how about degenerate conics?

How can I find the focii and directrices from the standard equation?
 
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