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7.1: Introduction to Conics

  • Page ID
    80794
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    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.

    Screen Shot 2022-04-21 at 4.19.38 PM.png

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

    Screen Shot 2022-04-21 at 4.20.07 PM.png

    Tilting the plane ever so slightly produces an ellipse.

    Screen Shot 2022-04-21 at 4.20.51 PM.png

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

    Screen Shot 2022-04-21 at 4.21.30 PM.png

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

    Screen Shot 2022-04-21 at 4.22.09 PM.png

    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.

    Screen Shot 2022-04-21 at 4.25.30 PM.png

    Screen Shot 2022-04-21 at 4.25.51 PM.png

    Screen Shot 2022-04-21 at 4.26.32 PM.png

    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.


    This page titled 7.1: Introduction to Conics is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Carl Stitz & Jeff Zeager via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.