c10s6 - 10.6 Conic Sections (Some of the following...

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10.6 Conic Sections (Some of the following information and figures are from the web site http://www.mathacademy.com/platonic_realms/encyclop/articles/conics.html ) Conic sections are the curves, which result from the intersection of a plane with a cone. These curves were studied and revered by the ancient Greeks, and were written about extensively by both Euclid and Appolonius. They remain important today, partly for their many and diverse applications. Now, in intersecting a flat plane with a cone, we have three choices, depending on the angle the plane makes to the vertical axis of the cone. First, we may choose our plane to have a greater angle to the vertical than does the generator of the cone, in which case the plane must cut right through one of the nappes. This results in a closed curve called an ellipse . Second, our plane may have exactly the same angle to the vertical axis as the generator of the cone, so that it is parallel to the side of the cone. The resulting open curve is called a parabola . Finally, the plane may have a smaller angle to the vertical axis (that is, the plane is steeper than the generator), in which case the plane will cut both nappes of the cone. The resulting curve is called a hyperbola , and has two disjoint “branches”. PARABOLA The set of all points in the plane whose distances from a fixed point F, called the focus , and a fixed line, called the directrix , are always equal. The vertex is the point halfway between the focus and the directrix.
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c10s6 - 10.6 Conic Sections (Some of the following...

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