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From: berry@zinfandel.UUCP (Berry Kercheval)
Newsgroups: net.math
Subject: Re: Question came up at a party
Message-ID: <490@zinfandel.UUCP>
Date: Thu, 23-Jan-86 18:40:41 EST
Article-I.D.: zinfande.490
Posted: Thu Jan 23 18:40:41 1986
Date-Received: Tue, 28-Jan-86 06:19:15 EST
References: <532@well.UUCP>
Reply-To: berry@zinfandel.UUCP (Berry Kercheval)
Distribution: net
Organization: Zehntel Inc., Walnut Creek CA
Lines: 66
Keywords: conic sections, general formula, analytic geometry
Summary: 
Summary: Yes, Virginia, there is a general formula.

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?

Yes.  My reference is the CRC Standard Mathematical Tables, 22 ed., page 373.

Consider the graph of

          2            2
	ax  + 2hxy + by  + 2gx +2fy + c = 0




               a  h  g                 a   h
let del = det  h  b  f ,    J = det           ,
               g  f  c                 h   b


                        a   g             b   f
I = a + B, K  =  det             + det              .
                        g   c             f   c

Then we can build this neat table:

Case   del   J    del/I    K     Conic
-----------------------------------------------
1      !=0   >0    <0            real ellipse  (circle is degenerate ellipse)
2      !=0   >0    >0            imag. ellipse
3      !=0   <0                  hyperbola
4      !=0   0                   parabola
5       0    <0                  real intersecting lines
6       0    >0                  conjugate complex lines
7       0    0             <0    real distinct parallel lines
8       0    0             >0    conjugate complex parallel lines
9       0    0              0    coincident lines


in cases 1, 2 and 3 the center (x0, y0) of the conic is given by the 
simultaneous solution of

	ax + hy + g = 0,    hx + by + f = 0

The equations of the axes of the conic are:

					
	y - y0 = m(x - x0),  y - y0 = - 1/m (x - x0),

where m is the positive root of
 
	   2
	 hm   + (a - b)m -h = 0.


I do not promis that I have made no errors of transcription, for a 
(more or less) guaranteed correct version consult the CRC std. math. tables;
it's a wondrous useful book!




-- 
Berry Kercheval		Zehntel Inc.	(ihnp4!zehntel!zinfandel!berry)
(415)932-6900				(kerch@lll-tis.ARPA)