How to determine a circle

Determining a circle means finding its three parameters: the centre (two coordinates) and the radius, or equivalently the coefficients $a$, $b$, $c$ of the standard form. Three independent conditions are therefore needed, each of which becomes an equation.

In summary — Methods

• Centre and radius known: apply directly the “vertex” form $(x-x_0)^2 + (y-y_0)^2 = R^2$.
• Three points: substitute them into the standard form $x^2 + y^2 + ax + by + c = 0$, obtaining a linear system of three equations in $a, b, c$.
• Mixed conditions: (tangent to one or two lines, passing through a point, fixed radius, centre on a line…) translate each condition into an equation in $a, b, c$ (or in $x_0, y_0, R$) and solve the system.

The advantage of the standard form in the three-point case is that the system turns out to be linear in $a$, $b$, $c$: the $x^2+y^2$ terms become known numbers once the coordinates are substituted.

Links

Topics: Circonferenza analitica
Concepts: Condizioni geometriche · Equazione circonferenza · Forma standard · Forma vertice
Methods: Circonferenza per tre punti
Skills: Geometria analitica · Risolvere sistemi