General Form of the Equation

Key Questions

• The general form of a circle looks like ...

${x}^{2} + {y}^{2} + A x + B y + C = 0$

In the standard form to the equation for a circle look like ...

${\left(x - h\right)}^{2} + {\left(y - k\right)}^{2} = {r}^{2}$

$\sqrt{{r}^{2}} = r ,$radius

Convert the general form to standard form by using the completing the square process.

You will then have the ${r}^{2}$ value.

The square root of ${r}^{2}$ is the radius of the circle.

• General equation of ellipse or circle

${\left(x - h\right)}^{2} / {a}^{2} + {\left(y - k\right)}^{2} / {b}^{2} = 1$

If $a = b$ then you have a circle.

If $a > b$ then you have an ellipse where the $x$ axis is the major axis.

If $b > a$ then you have an ellipse where the $y$ axis is the major axis.

A circle in general form has the same non-zero coefficients for the ${x}^{2}$ and the ${y}^{2}$ terms. So if there is a graph, it is a circle (or a point).

Explanation:

Don't be too hasty, though.

$A {x}^{2} + B x y + C {y}^{2} + D x + E y + F = 0$

Assuming that there is ineed a graph, it is:

an ellipse if $A$ and $C$ have the same sign.

a circle if $A = C$.

However it is possible that there is no graph:

${x}^{2} + {y}^{2} = - \frac{9}{4}$ Has no graph, but it can be rewritten as:

$4 {x}^{2} + 4 {y}^{2} + 9 = 0$. At first look, this appears to be the equation of a circle, but it is not.