Conic sections
A circle, an oval, a dish-shaped curve and a curve in two pieces look unrelated. What do they have in common?
Where you are 6 inside
What this area is about
Cut a double cone with a flat plane and, when the plane misses the vertex, the cut is a circle, an ellipse, a parabola or a hyperbola, depending only on its tilt. These four curves are the conic sections. Each can also be described by distances in the plane, and that description turns it into an equation in x and y.
How these fit together
The family. Conic sections from a double cone shows all four as slices of one solid. Focus and directrix gives one planar rule for the three non-circles: the points with PF = e \cdot PL, where e < 1 gives an ellipse, e = 1 a parabola and e > 1 a hyperbola.
One curve at a time. The ellipse keeps the sum of distances to two foci fixed, the hyperbola keeps their difference fixed, and the parabola as a locus keeps a point equally far from a focus and a directrix. Each gets its standard equation.
The procedure. Identifying and translating conics reads an expanded equation Ax^2 + Cy^2 + Dx + Ey + F = 0, with A and C not both 0, and says which curve it is and where it sits.
A path through it
Two strands meet at the end. One runs from the double cone to focus and directrix, then to the parabola as a locus. The other runs from the ellipse to the hyperbola. Identifying conics comes last, after both the parabola and the hyperbola.
The circle itself is studied in the circles area, and the identifying lesson uses its equation as well.