8.1 Introduction to Conics

In this chapter, we study the Conic Sections - literally `sections of a cone'. Imagine a double-napped cone as seen below being `sliced' by a plane.

A closed oval sits to the right of a marked point at the origin with a dashed green vertical line beyond it, and a slider labeled with the eccentricity reshapes the curve continuously: the oval stretches rightward, opens into a parabola at the value one, and then splits into two opposite branches that sweep out of view for larger values. A second slider moves the dashed vertical line and scales the whole curve with it, while the marked focus stays fixed at the origin throughout. Adjustable parameters: Eccentricity e (e) = 0.6, Directrix distance d (d) = 2. Viewing window: x from -13.62 to 15.62, y from -9.04 to 9.04. — Fix a focus at the origin and a vertical directrix x = d. The set of points whose distance to the focus is e times their distance to the directrix is r = ed/(1 + e cos θ) in polar form, and the single dial e runs through the whole family: an ellipse for e < 1, a parabola at exactly e = 1, and a hyperbola for e > 1, whose two branches appear as the curve shoots off toward the directions where 1 + e cos θ vanishes. This is the same family the slicing plane produces from the cone, described without the cone.

If we slice the cone with a horizontal plane the resulting curve is a circle.

Tilting the plane ever so slightly produces an ellipse.

If the plane cuts parallel to the cone, we get a parabola.

If we slice the cone with a vertical plane, we get a hyperbola.

If the slicing plane contains the vertex of the cone, we get the so-called `degenerate' conics: a point, a line, or two intersecting lines.

While this geometric introduction to the conic sections has its uses,¹ in order to study the applications of the conic sections, we require a more analytic approach. It turns out each of the of the conic sections can be described as as a locus of points - that is, a set of points which satisfy a certain condition involving distance. The reader is referred to Section A.3 for a review of the distance and related formulas.

As we'll see, we'll be able to use the distance formula to algebraically represent the conic sections as graphs of general quadratic equations in two variables. That is, every conic section in the $x y$ -plane can be represented as the graph of an equation of the form $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ for real numbers $A$ , $B$ , $C$ , $D$ , $E$ , and $F$ .

Adapted from Precalculus with Integrated Calculus, edition 4i, by Carl Stitz and Jeff Zeager (stitz-zeager.com), licensed under CC BY-NC-SA 3.0. Changes were made: reformatted as an accessible XYZ web edition. License: CC-BY-NC-SA-3.0.