General Form of the Equation of a Circle

Consider the equation of a 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.

If we rearrange the terms of the above equation (i) of a 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 a circle with the standard equation of a circle {\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2} and 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 a circle.

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

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

Solution: The given equation of a circle is

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

We observe that in this equation of a circle the coefficients of {x^2} and {y^2} is 7, but in the general form of the equation of a circle the coefficients must be equal to 1.

To convert the given equation into the form of general equation, divide the given equation on both sides by 7. We get

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

The above equation can 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 a 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)

The 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}