# General Form of the Equation of a Circle

Consider the equation of a circle in general form is

Where $g,f,c$ are any constant values.

If we rearrange the terms of the above equation (i) of a circle, we have

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.:

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

Solution: The given equation of a circle is

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

The above equation can be written as

Compare this equation with the general equation of a circle as

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 $r = \sqrt {{g^2} + {f^2} - c} = \sqrt {{{\left( { - \frac{9}{7}} \right)}^2} + {{\left( {\frac{5}{7}} \right)}^2} - 2} = \frac{{2\sqrt 2 }}{7}$