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6: Conic Sections

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    In this chapter, we study the Conic Sections - literally 'sections of a cone'.

    • 6.1: Introduction to Conics
      In this chapter, we study the Conic Sections - literally `sections of a cone'.  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.
    • 6.2: Circles
      Recall from Geometry that a circle can be determined by fixing a point (called the center) and a positive number (called the radius) as follows.
    • 6.3: Ellipses
      We may imagine taking a length of string and anchoring it to two points on a piece of paper. The curve traced out by taking a pencil and moving it so the string is always taut is an ellipse.
    • 6.4: Parabolas
      We have already learned that the graph of a quadratic function is called a parabola. To our surprise and delight, we may also define parabolas in terms of distance in conics.
    • 6.5: Hyperbolas
      In the definition of an ellipse, we fixed two points called foci and looked at points whose distances to the foci always added to a constant distance \(d\). Those prone to syntactical tinkering may wonder what, if any, curve we'd generate if we replaced added with subtracted. The answer is a hyperbola.

    Thumbnail: The circle and ellipse conic sections is determined by the angle the plane makes with the axis of the cone. The other two conic section (the hyperbola and parabolas) are not shown. (CC BY-SA 4.0; OpenStax)

    This page titled 6: Conic Sections 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.