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From: batu@hound.UUCP (S.PETERSON)
Newsgroups: net.math
Subject: Re:Re: Question came up at a party
Message-ID: <1622@hound.UUCP>
Date: Thu, 23-Jan-86 12:25:26 EST
Article-I.D.: hound.1622
Posted: Thu Jan 23 12:25:26 1986
Date-Received: Fri, 24-Jan-86 09:04:00 EST
References: <532@well.UUCP>
Organization: AT&T Bell Labs, Holmdel NJ
Lines: 34

The question is a "formula" for "all" the conic sections


For simplicity assume that they are in R**2 and centered at the origin,
 (note to have centers not at the origin is a simply translation)

Let a & b be nonzero real numbers, r a non negative real number, x and y 
unknowns as usual.

Then we have 

	2	2
(P)   ax    + by  = r

i) if a=b and both are > 0 the P is a circle.

ii) if a is not equal to b and both are > 0 the P is an ellipse, and there 
	is a formula for finding axes, focii etc.

iii) if a is not b and one is > 0, the other < 0, then we have a hyperbola,
	again there are formulas (in terms of a,b and r) for asymptopes, etc.
				  2               2
(Q) to get a parabola, e.g. y = ax   or     x = by


In general an equation of the form 

     2                       2
   ax   + bx  + cxy + dy + ey  = r

gives a conic where a,b,c,d,e,r are any real numbers.
( if b=c=e=r=0, then we have a point, which is a conic section)
Consult any High school algebra text or a college Analyitical geometry text.