This free online calculator finds the center and radius of a circle given its equation in general form:
\[ x^2 + y^2 + ax + by = c \]The calculator converts this equation into standard form by completing the square.
Group the \(x\) and \(y\) terms:
\[ x^2 + ax = \left(x+\frac{a}{2}\right)^2 - \left(\frac{a}{2}\right)^2 \] \[ y^2 + by = \left(y+\frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 \]Substitute these expressions into the original equation to obtain the standard form:
\[ (x-h)^2 + (y-k)^2 = r^2 \]where
\[ (h,k)=\left(-\frac{a}{2},-\frac{b}{2}\right) \]and
\[ r^2 = c + \left(\frac{a}{2}\right)^2 + \left(\frac{b}{2}\right)^2 \]So the center is \( (h,k) \) and the radius is \( r \).
Enter the coefficients \(a\), \(b\), and \(c\), choose the number of decimal places, then click Enter. If the equation represents a circle, the calculator will display the center coordinates and radius.
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