anonymous
  • anonymous
What is the standard equation of a circle?
Mathematics
katieb
  • katieb
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Michele_Laino
  • Michele_Laino
here is the standard equation of a circle: \[\Large {x^2} + {y^2} + ax + by + c = 0\] where, a, b, and c are real coefficient, which satisfy that condition: \[\Large \frac{{{a^2}}}{4} + \frac{{{b^2}}}{4} - c > 0\]
Michele_Laino
  • Michele_Laino
coefficients*
anonymous
  • anonymous
I always thought the "standard form" of the equation of a circle was\[\left( x-a \right)^{2} + \left( y-b \right)^{2} = r ^{2}\]where the center of the circle is at (a, b) and the radius is r.

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anonymous
  • anonymous
Haha, me too...
freckles
  • freckles
I think the first way mentioned by Michele is the general form and the way mentioned by osprey is the standard form. Though I'm not sure if these terms hold universal.
anonymous
  • anonymous
@Michele_Laino 's equation results in an ellipse.
anonymous
  • anonymous
Ok, think I got the answer, thanks!
Michele_Laino
  • Michele_Laino
please, note that, my equation is not the equation of an ellipse @ospreytriple
anonymous
  • anonymous
Let a=4, b=3, and c=1, for example. Graph it and see what you get, @Michele_Laino
Michele_Laino
  • Michele_Laino
that is a condition, on parameters a, b, and c
anonymous
  • anonymous
???? If the equation truly represents a circle, then the values of a, b, and c shouldn't matter so long as they satisfy the condition you mentioned. It should always produce a circle. And it doesn't.
freckles
  • freckles
Michele_Laino has written general form for equation @ospreytriple \[x^2+y^2+ax+by+c=0 \\ x^2+ax+y^2+by=-c \\ (x^2+ax+(\frac{a}{2})^2)+(y^2+by+(\frac{b}{2})^2)=-c +(\frac{a}{2})^2+(\frac{b}{2})^2 \\ (x+\frac{a}{2})^2+(y+\frac{b}{2})^2 =\frac{a^2}{4}+\frac{b^2}{4}-c \\ \text{ center is } (\frac{-a}{2},\frac{-b}{2}) \\ \text{ and radius is } \sqrt{\frac{a^2}{4}+\frac{b^2}{4}-c} \\ \text{ this is a circle if } \frac{a^2}{4}+\frac{b^2}{4}-c>0 \]
freckles
  • freckles
general form for circle *
freckles
  • freckles
http://www.wolframalpha.com/input/?i=x%5E2%2By%5E2%2B4x%2B3y%2B1%3D0 the example you picked shows a circle
anonymous
  • anonymous
Must be something wrong with my version of Autograph as it clearly shows an ellipse. If so, please accept my apologies. I do, however, stand by my "standard form" equation.
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