From the definition to the equation

We derive the equation of the circle starting from its locus definition, translating the condition $d(P,C)=R$ into coordinates.

Proof

From the locus definition: $\sqrt{(x-x_0)^2 + (y-y_0)^2} = R$. The two sides are $\ge 0$ ($R>0$ and the first is a distance), so we can square without further conditions, obtaining the “vertex” form. Expanding the squares and rearranging: $x^2 - 2x_0 x + x_0^2 + y^2 - 2y_0 y + y_0^2 = R^2$ which, with the settings $a=-2x_0$, $b=-2y_0$, $c=x_0^2+y_0^2-R^2$, becomes the standard form. Inverting the formulae is immediate. ∎

The key point is that squaring introduces no extraneous solutions, because both sides of the equality are already non-negative.

Links

Topics: Analytical circle
Concepts: Distance between points · Circle equation · Standard form · Vertex form · Cartesian plane
Skills: Proving · Analytical geometry