General and standard form of a circle

A circle's equation is sometimes given in an expanded general form. From $x^2 + y^2 - 6x + 4y - 3 = 0$ we convert to the standard form, from which the centre and radius can be read off¹.

Completing the square

We complete the square in $x$ and in $y$ separately, using $x^2 - 6x = (x - 3)^2 - 9$ and $y^2 + 4y = (y + 2)^2 - 4$ .

\begin{align*} (x - 3)^2 - 9 + (y + 2)^2 - 4 - 3 &= 0 \\ (x - 3)^2 + (y + 2)^2 &= 16 \end{align*}

Comparing with the standard form, the centre is $(3, -2)$ and the radius is $4$ .

Why the standard form reads so easily

It is the definition of a circle written out: the set of points at distance $r$ from the centre. Squaring both sides of the distance formula $\sqrt{(x - a)^2 + (y - b)^2} = r$ produces it directly.

Half of the $x$ -coefficient $-6$ is $-3$ , and half of the $y$ -coefficient $4$ is $2$ ; these are the centre coordinates with signs flipped.

Not every such equation is a circle

After completing the square, the right-hand side decides everything.

Constant term	Right-hand side	The locus
$-3$	$16$	a circle of radius $4$
$+13$	$0$	the single point $(3, -2)$
$+14$	$-1$	no real points at all

A sum of squares is never negative, so the last case has no solution.

Two further conditions

The coefficients of $x^2$ and $y^2$ must be equal
There must be no $xy$ term

Unequal coefficients give an ellipse, and an $xy$ term tilts the figure.

The large dots on the graph are the centre and a point on the circle, a radius to its right.

Completing the square, Wikipedia