9. The Circle.—It is easy to write down the equation of a given circle. Let (h, k) be its given centre C, and ρ the numerical measure of its given radius. Take P (x, y) any point on its circumference, and construct the triangle CRP, in fig. 50 as above. The fact that this is right-angled tells us that
CR² + RP² = CP²,
and this at once gives the equation
(x − h)² + (y − k)² = ρ².
A point not upon the circumference of the particular circle is at some distance from (h, k) different from ρ, and satisfies an equation inconsistent with this one; which accordingly represents the circumference, or, as we say, the circle.
The equation is of the form
x² + y² + 2Ax + 2By + C = 0.
Conversely every equation of this form represents a circle: we have only to take −A, −B, A² + B² − C for h, k, ρ² respectively, to obtain its centre and radius. But this statement must appear too unrestricted. Ought we not to require A² + B² − C to be positive? Certainly, if by circle we are only to mean the visible round circumference of the geometrical definition. Yet, analytically, we contemplate altogether imaginary circles, for which ρ² is negative, and circles, for which ρ = 0, with all their reality condensed into their centres. Even when ρ² is positive, so that a visible round circumference exists, we do not regard this as constituting the whole of the circle. Giving to x any value whatever in (x − h)² + (y − k)² = ρ², we obtain two values of y, real, coincident or imaginary, each of which goes with the abscissa x as the ordinate of a point, real or imaginary, on what is represented by the equation of the circle.
The doctrine of the imaginary on a circle, and in geometry generally, is of purely algebraical inception; but it has been in its entirety accepted by modern pure geometers, and signal success has attended the efforts of those who, like K.G.C. von Staudt, have striven to base its conclusions on principles not at all algebraical in form, though of course cognate to those adopted in introducing the imaginary into algebra.
A circle with its centre at the origin has an equation x² + y² = ρ².