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 November 13th, 2010, 12:36 PM #1 Senior Member   Joined: Apr 2009 Posts: 106 Thanks: 0 Circle Problem Find a necessary and sufficient condition on A, B, C, D, and E for Ax^2 + By^2 + Cx + Dy + E = 0 to be an equation of a circle.
November 13th, 2010, 03:57 PM   #2
Math Team

Joined: Dec 2006
From: Lexington, MA

Posts: 3,267
Thanks: 408

Re: Circle Problem

Hello, julian21!

I'll give this a try . . .

Quote:
 $\text{Find a necessary and sufficient condition on }A,\,B,\,C,\,D,\text{ and }E$ [color=beige]. . [/color]$\text{ for }Ax^2\,+\,By^2\,+\,Cx\,+\,Dy\,+\,E\:=\:0\,\text{ to be an equation of a circle.}$

$\text{First of all: }\:A \,=\,B.$

$\text{We have: }\:Ax^2\,+\,Cx\,+\,Ay^2\,+\,Dy \:=\:-E$

$\text{Divide by }A:\;\;x^2\,+\,\frac{C}{A}x\,+\,y^2\,+\,\frac{D}{A }y \;=\;-\frac{E}{A}$

$\text{Complete the square: }\;x^2\,+\,\frac{C}{A}x\,+\,\frac{C^2}{4A^2}\,+\,y ^2\,+\,\frac{D}{A}y\,+\,\frac{D^2}{4A^2} \;=\;\frac{C^2}{4A^2}\,+\,\frac{D^2}{4A^2}\,-\,\frac{E}{A}$

$\text{And we have: }\;\left(x\,+\,\frac{C}{2A}\right)^2\,+\,\left(y\, +\,\frac{D}{2A}\right)^2 \;=\;\frac{C^2\,+\,D^2\,-\,4AE}{4A^2}$

$\text{The circle has center: }\,\left(-\frac{C}{2A},\:-\frac{D}{2A}\right)\:\text{ and radius: }\:r \;=\;\frac{\sqrt{C^2\,+\,D^2\,-\,4AE}}{2A},$

[color=beige]. . [/color]$\text{provided that: }\:A\,\ne\,0,\;A \,=\,B,\;\text{ and }\:C^2\,+\,D^2\:\ge\:4AE$

 November 15th, 2010, 11:54 AM #3 Senior Member   Joined: Apr 2009 Posts: 106 Thanks: 0 Re: Circle Problem Beautiful!!! my only question is... why does A = B, or are we assuming this?
 November 15th, 2010, 12:09 PM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Circle Problem In order for the equation to describe a circle A = B is a requirement. If you take one of the general equations for a circle in the plane: $(x-h)^2+(y-k)^2=r^2$ Expanding the squared binomials gives: $x^2-2hx+h^2+y^2-2ky+k^2=r^2$ Now, multiplying through by any non-zero constant will still result in the coefficients of x² and y² being equal.

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