the circle 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. The ellipse 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.