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I'm writing a paper on conic sections for my math research class in high school, and I was wondering if any of you know any challenging/interesting topics I can include in my paper regarding conic sections. I already have the basics i.e. proofs describing eccentricity and foci, but what else should I include? Below, I'm going to post some links to some questions regarding conic sections I found on Math.SE, and I would really appreciate it if I can get some of your opinions on whether or not I should include those topics. Thanks!

Usefulness of Conic Sections http://www.fen.bilkent.edu.tr/~franz/publ/conics.pdf The intersection of two conics - matrix solution CONICS, computing intersections, passing to complex numbers Visualizing the flat complex conic Rank of a degenerate conic Solving a Conic Matrix given these Equations Show that the rational conic $F(x,y)=ax^2+bxy+cy^2+dx+ey+f$, subject to a certain condition, is non-singular

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I'm not entirely sure what the level of the paper should be, but for a high school project, I think a thorough description of all possible conic sections and why they're all possibilities (including degenerate ones like a point or a pair of lines) would be quite ok already. I would focus on 1) What are all conic sections and why, 2) Why are some called degenerate? and 3) explicitly describing some as you already said you did. –  HSN Jan 22 '13 at 0:30
Yes, I've already done that, but I'm trying to go beyond regular high school mathematics because I want to have the best paper in my county (my school's math research class enters this county-wide math fair every year). –  joejacobz Jan 22 '13 at 1:05
In that case, I think the last question is one of interest, indeed: when the equation of a conic section is known, how do we know it is (non-)singular? Another thing that might really help you, is Pascal's theorem: given six points $P_1,\ldots,P_6$, these give a hexangle. We can prolong its sides and consider the three intesrection points of `opposing' sides. These three points lie on one single line if and only if the six points lie on a conic section. For this you at least need to say something about projective geometry, since the line ay be at infinity (consider a regular hexagon!). –  HSN Jan 22 '13 at 1:21
@HSN What kind of math would it require besides projective geometry? –  joejacobz Jan 22 '13 at 3:01
That depends on the way you wish to prove it. There are several proofs out there and some of them are easier than others. Just search a bit and you'll find them. I personally like the proof using cubic curves (which could for instance be found in M. Reid - Undergraduate Algebraic Geometry), but that might be a bit far-fetched. –  HSN Jan 22 '13 at 16:09

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