7.1: Introduction to Conics

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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.