



 
Christine, Like many topics in Math, there are multiple ways to come at this. One could describe a cone, in 3space, and look at the sections of this by a plane (the name conic sections gives this story). To get all of them, you may have to explore several different cones, including the extreme case of a circular cylinder. You could look at the 'locus' definition  the set of points making the sum (ellipse) or difference (hyperbola) of the distances from a pair of points. There is yet another special case  the parabola (equal distances from a point and a line). However, you said this is an analysis course. There is yet another route into these: If you pick any five points in the plane (no more than three on a line) then there will be a unique curve of the form above which runs through all five points. (That is easiest seen using linear algebra in the coefficients.) It will be a conic section. Walter Whiteley  


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