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From: bs@linus.UUCP (Robert D. Silverman)
Newsgroups: net.math
Subject: Re: Question came up at a party
Message-ID: <144@linus.UUCP>
Date: Wed, 22-Jan-86 20:29:20 EST
Article-I.D.: linus.144
Posted: Wed Jan 22 20:29:20 1986
Date-Received: Fri, 24-Jan-86 21:31:55 EST
References: <532@well.UUCP>
Distribution: net
Organization: The MITRE Corporation, Bedford, MA
Lines: 39

> 
> 	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?
> 
> 	This came up at a party, when one person maintained that the
> various conic sections are fundamentally different, and the rest of us
> that they are all 'special cases' of the same thing.  Finally, he
> challenged us to write an equation as above.  Nobody could do it!
> (But then, nobody there had a degree in math [if that means anything].)
> 
> 	Please reply on net.math
> 
> 
>        Robert Bickford     (rab@well.uucp)

This is a simple high school analytic geometry problem. A general quadratic
form in two variables is:

		   2     2
	F(x,y) = Ax  + Cy  + Dx + Ey + F
 
F(x,y) = 0 is the general equation representing a conic section. The following
conditions hold:

If A or C is zero the conic is a parabola or in special cases two parallel
lines which may be distinct, coincident or imaginary.

If A and C have the same sign we have an ellipse or in special cases it can
degenerate into a single point or imaginary ellipse. It is a circle when
A = C.

If A and C have different signs we have a hyperbola or in degenerate cases
two intersecting lines.


I hope this is what you were looking for.

Bob Silverman