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📚 Precalculus
Chapters ▾
1 Relations and Functions2 Linear and Quadratic Functions3 Polynomial Functions4 Rational Functions5 Further Topics in Functions6 Exponential and Logarithmic Functions7 Hooked on Conics8 Systems of Equations and Matrices9 Sequences and the Binomial Theorem10 Foundations of Trigonometry11 Applications of Trigonometry

7.1 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 7.1

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

Image: Circle01
Figure 7.2
Image: Circle02
Figure 7.3

Tilting the plane ever so slightly produces an ellipse.

Image: Ellipse01
Figure 7.4
Image: Ellipse02
Figure 7.5

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

Image: Parabola01
Figure 7.6
Image: Parabola02
Figure 7.7

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

Image: Hyperbola01
Figure 7.8
Image: Hyperbola02
Figure 7.9

For a wonderful animation describing the conics as intersections of planes and cones, see Dr. Louis Talman's Mathematics Animated Website .

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 7.10
Image: Point02
Figure 7.11
Image: Ilines01
Figure 7.12
Image: Ilines02
Figure 7.13
Image: Tline01
Figure 7.14
Image: Tline02
Figure 7.15

We will focus the discussion on the non-degenerate cases: circles, parabolas, ellipses, and hyperbolas, in that order. To determine equations which describe these curves, we will make use of their definitions in terms of distances.

Adapted from Precalculus, 3rd corrected edition, 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.