General Form of Equation of Circle

Consider the equation of circle in general form is

\boxed{{x^2}  + {y^2} + 2gx + 2fy + c = 0}\,\,\,\,\,\,{\text{ -  -  -  }}\left( {\text{i}} \right)


Where g,f,c are any constant values.
Rearrange the terms of the above equation (i) of circle, we have

{x^2}  + 2gx + {y^2} + 2fy + c = 0\,\,\,\,\,\,{\text{ -  -  -  }}\left( {{\text{ii}}} \right)


In this equation we use the method of completing squares, so for this we need to add {g^2} and {f^2} on both sides of the equation (ii). i.e.

\begin{gathered} {x^2} + 2gx = {g^2} + {y^2} + 2fy + {f^2} + c  = {g^2} + {f^2} \\ \Rightarrow {\left( {x + g} \right)^2} +  {\left( {y + f} \right)^2} = {g^2} + {f^2} - c \\ \Rightarrow {\left[ {x - \left( { - g}  \right)} \right]^2} + {\left[ {y - \left( { - f} \right)} \right]^2} = {\left[  {\sqrt {{g^2} + {f^2} - c} } \right]^2} \\ \end{gathered}


Compare this equation of circle with standard equation of circle {\left( {x - h} \right)^2} + {\left(  {y - k} \right)^2} = {r^2} we get the radius \sqrt {{g^2} + {f^2} - c} and centre \left( { - g, - f} \right).
This shows that the equation {x^2} +  {y^2} + 2gx + 2fy + c = 0 represents a circle with centre \left( { - g, - f} \right) and radius \sqrt {{g^2} + {f^2} - c} . This is called the general equation of circle.

Example: Find the centre and radius of the circle with the given equation of circle

7{x^2}  + 7{y^2} + 18x - 10y + 14 = 0


Solution: We have given equation of circle is

7{x^2}  + 7{y^2} + 18x - 10y + 14 = 0


We observe that in this equation of circle the coefficients of {x^2} and {y^2} is 7, but in the general form of equation of circle coefficients must be equal to 1.
To convert the given equation in form of general equation, dividing given equation both sides by 7, we get

{x^2}  + {y^2} + \frac{{18}}{7}x - \frac{{10}}{7}y + 2 = 0


The above equation cam be written as

{x^2}  + {y^2} + 2\left( {\frac{9}{7}} \right)x + 2\left( { - \frac{5}{7}} \right)y +  2 = 0


Compare this equation with the general equation of circle as

{x^2}  + {y^2} + 2gx + 2fy + c = 0


We have the values g =  \frac{9}{7}, f =  - \frac{5}{7} and c = 2.

Hence the centre of the circle is \left( { - g, - f} \right) = \left( { -  \frac{9}{7}, - \left( { - \frac{5}{7}} \right)} \right) = \left( { -  \frac{9}{7},\frac{5}{7}} \right)

Radius of the circle is r  = \sqrt {{g^2} + {f^2} - c}  = \sqrt  {{{\left( { - \frac{9}{7}} \right)}^2} + {{\left( {\frac{5}{7}} \right)}^2} -  2}  = \frac{{2\sqrt 2 }}{7}

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