Implicit and explicit form

The equation of a circle can be written in two equivalent forms: the “vertex” form, which highlights centre and radius, and the “standard” form, which is a second-degree equation with $x^2$ and $y^2$ terms having equal coefficients.

Property — Implicit and explicit form

The equation of a circle with centre $C(x_0;y_0)$ and radius $R$ is $(x - x_0)^2 + (y - y_0)^2 = R^2 \qquad \text{(forma ``vertice'')}$ which, on expanding, becomes $x^2 + y^2 + a x + b y + c = 0 \qquad \text{(forma ``standard'')}$ with the correspondence $a = -2 x_0,\quad b = -2 y_0,\quad c = x_0^2 + y_0^2 - R^2.$ Inverting, given the equation in standard form the centre and radius are recovered: $\boxed{C\left(-\tfrac{a}{2};\ -\tfrac{b}{2}\right),\qquad R^2 = \tfrac{a^2}{4} + \tfrac{b^2}{4} - c.}$

The two forms are used in different situations: the vertex form is convenient when centre and radius are known, the standard form when the circle is given by an equation or is to be determined by imposing conditions.

Links

Topics: Analytical circle
Concepts: Centre · Circle equation · Standard form · Vertex form · Radius
Methods: Circle equation
Skills: Analytical geometry · Using formulae