c10s6

# c10s6 - 10.6 Conic Sections(So m e ofthe following...

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10.6 Conic Sections (Som e  of the  following  information  and  figures  are  from  the  web  site   http://www.mathacad e m y.c o m/platonic_realms/encyclop/articles/conics.html ) Conic  sections  are  the  curves,  which  result from  the  intersection  of a  plane  with a  cone.  Thes e  curves  were  st   the  ancient  Greeks,  and  were  written  about  extensively  by  both  Euclid  and  Appolonius.   They   remain   important   today,   partly   for   their   many   and   diverse   applicatio   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  distance s  from  a  fixed  point  F, called  the   focus , and  a  fixed   line,  called  the

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c10s6 - 10.6 Conic Sections(So m e ofthe following...

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