Conic Sections and the Locus Definition of a Parabola

Slice a double cone with a plane and, depending on the angle of the cut, you get a circle, an ellipse, a parabola, or a hyperbola — the four conic sections. If the plane is tilted at exactly the same angle as the cone's own slant (its generating angle), the cross-section that results is a parabola: an open, U-shaped curve, unbounded in one direction. This is also the shape a ball traces when you throw it (ignoring air resistance), which is where the Greek-derived name comes from.

While the cone picture explains where the curve comes from, it is awkward to compute with. For algebra, it is far more useful to define a conic as a locus — a set of points satisfying a distance rule. Pick a fixed point $S$ (the focus) and a fixed line $l$ (the directrix) not passing through $S$. For any point $P$ in the plane, let $PM$ be the perpendicular distance from $P$ to $l$. The locus of every point $P$ for which the ratio

is called a conic, and $e$ is its eccentricity. The line through the focus perpendicular to the directrix is the axis of the conic. Different values of $e$ produce different curves — and this single number is what tells the three main conics apart:

• $e = 1$: the conic is a parabola — the case this chapter is entirely about.
• $0 < e < 1$: the conic is an ellipse.
• $e > 1$: the conic is a hyperbola.

So a parabola, in the language we will use throughout, is simply the set of all points that are exactly as far from a fixed focus as they are from a fixed directrix. Nothing about a cone is needed to work with it algebraically — the distance rule with $e=1$ is the entire definition, and every property of the parabola (its equation, its tangents, its normals) is eventually squeezed out of that one equality of distances.

Worked example. A point $P$ moves so that its distance from the point $S(4,0)$ always equals its perpendicular distance from the line $x=-4$. What kind of curve does $P$ trace, and what is its eccentricity? …