Precalculus with Integrated CalculusXYZ Homework Edition

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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.

Image: cone
Figure 8.1

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

Image: Circle01
Figure 8.2
Image: Circle02
Figure 8.3

Tilting the plane ever so slightly produces an ellipse.

Image: Ellipse01
Figure 8.4
Image: Ellipse02
Figure 8.5

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

Image: Parabola01
Figure 8.6
Image: Parabola02
Figure 8.7

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

Image: Hyperbola01
Figure 8.8
Image: Hyperbola02
Figure 8.9

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.

Image: Point01
Figure 8.10
Image: Point02
Figure 8.11
Image: Ilines01
Figure 8.12
Image: Ilines02
Figure 8.13
Image: Tline01
Figure 8.14
Image: Tline02
Figure 8.15

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, Preliminary 4th Edition (integrated calculus), 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.

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