PrecalculusXYZ Homework Edition

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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.

Interactive figureEvery conic from one rule: distance to the focus is e times distance to the directrixDrag the Eccentricity e and Directrix distance d sliders.
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.
XYZ Graph · viewer build 5edf91b
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.
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.

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