A circle with centre O(0,0) and radius r has equation x2 + y2 = r2, and so
is included in the list of loci with equations quadratic in x and y. It is not
a degenerate conic as we defined them. It also occurs as a section of a
right-circular cone - take x2 + y2 = z2
cut by z = r. It does not, however,
appear in the list of loci with the focus-directrix property.
To see this, rewrite the equation as x2/r2 + y2/r2 = 1.
This looks like the
equation of an ellipse with a = b = r. But then we should have the relation
b2 = a2(1-e2), which implies that e = 0. We can think of this as the limiting
position of the ellipse x2/r2 + y2/r2(1-e2) = 1 as e tends to 0.
has focus (ae,0) and directrix x = a/e. As e tends to zero, the focus tends
to (0,0) - the centre of both the ellipse and the circle. The directrix has no
meaningful limiting position.
We can also see that the circle is anomalous by considering its symmetry
group. It is easy to see that the circle C : x2 + y2 = 1 has symmetry group
consisting of the rotations about O(0,0), and reflections in lines through O.
This group is infinite, whereas plane conics have symmetry groups of order
two or four.