A circle is the set of all points in a plane that lie at a fixed distance from a fixed point. The fixed point is the centre and the fixed distance is the radius. Everything you need about a circle follows from this single sentence applied through the distance formula.
Standard equation: if the centre is $(h, k)$ and the radius is $r$, a point $(x, y)$ lies on the circle exactly when its distance from the centre equals $r$. Squaring the distance formula gives:
When the centre is the origin $(0, 0)$ this collapses to the form you will use most often:
General form: expanding the standard equation and renaming constants gives the second-degree equation in which a circle is usually hidden:
Comparing this with the expanded standard form, the centre is $(-g, -f)$ and the radius is $r = \sqrt{g^2 + f^2 - c}$. Notice the two telltale signs of a circle inside a general second-degree equation: the coefficients of $x^2$ and $y^2$ are equal, and there is no $xy$ term.
| Form | Equation | Centre | Radius |
|---|---|---|---|
| Centre $(h,k)$ | $(x-h)^2+(y-k)^2=r^2$ | $(h,k)$ | $r$ |
| Centre at origin | $x^2+y^2=r^2$ | $(0,0)$ | $r$ |
| General form | $x^2+y^2+2gx+2fy+c=0$ | $(-g,-f)$ | $\sqrt{g^2+f^2-c}$ |
The quantity under the root, $g^2 + f^2 - c$, decides whether the equation even represents a real circle: if it is positive you have a genuine circle, if zero the "circle" shrinks to the single point $(-g, -f)$, and if negative no real point satisfies the equation at all.
Deeper Insight — every circle equation is just the distance formula in disguise: The reason a circle has such a tidy equation is that its definition is purely metric — it is built from one constant distance. When you read off the centre as $(-g, -f)$ from the general form, you are really completing the square twice, once in $x$ and once in $y$, to recover the perfect-square brackets that the squared distance always produces. This is why the technique of completing the square is the master skill for the whole chapter: it converts any messy general equation back into the geometry-revealing standard form. The condition $g^2 + f^2 - c > 0$ is not an arbitrary rule either; it simply demands that the squared radius be positive, because no real distance can be the square root of a negative number. Once you see the equation as "squared distance equals squared radius", finding centres and radii stops being a formula hunt and becomes a single, reliable manoeuvre.