Christine,

Like many topics in Math, there are multiple ways to come at this.

One could describe a cone, in 3-space, 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.
This suggests that they may be looking for something about the algebraic form.
Look for the properties of quadratic formula in x and y combined (i.e. x^2, y^2, xy, x, y, constant) and see what forms those take as curves in the plane.

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