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From: lhl@lanl.ARPA
Newsgroups: net.math
Subject: Re: Question came up at a party
Message-ID: <184@lanl.ARPA>
Date: Thu, 23-Jan-86 10:31:11 EST
Article-I.D.: lanl.184
Posted: Thu Jan 23 10:31:11 1986
Date-Received: Sat, 25-Jan-86 07:46:10 EST
References: <532@well.UUCP>
Reply-To: lhl@a.UUCP (Lewis Lowe)
Distribution: net
Organization: The New Mexico Institute for the Incurably Different
Lines: 37

In article <532@well.UUCP> rab@well.UUCP (Bob Bickford) writes:
>
>	Is there a formula which describes *all* conic sections, which
>will generate a particular class of same (e.g., circles, hyperbolas)
>when certain coefficients are plugged into it?
>
All conics have the form

      2          2
    Ax + Bxy + Cy  + Dx + Ey + F = 0

		       2
where either (A+C) or (B  - 4AC) is not equal to zero (this degenerate case
is either a line, the entire plane, or the null set.)

By rotation and translation of axes, all other combinations reduce to one of
two forms; either

      2
    Ax  + Ey + F = 0

with A nonzero; this is a parabola with E nonzero, otherwise (depending on
AF) a line, two parallel lines, or the null set.

The other case is

      2     2
    Ax  + Cy  + F = 0

with AC nonzero.  If AC < 0 this is a hyperbola or two intersecting lines,
depending on F.  If AC > 0 this is an ellipse, a point, or the null set,
depending on AF.

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He's got feet of clay clear up to his eyebrows.  A little of something else
at the very top though.                                         R. A. Lafferty