Questions about conic sections

[gangsta316]

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.

 

[HallsofIvy]

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".)

 

[gangsta316]

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?