the circle
A circle with centre O(0,0) and radius r has equation x^{2} + y^{2} = r^{2}, 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
rightcircular cone  take x^{2} + y^{2} = z^{2}
cut by z = r. It does not, however,
appear in the list of loci with the focusdirectrix property.
To see this, rewrite the equation as x^{2}/r^{2} + y^{2}/r^{2} = 1.
This looks like the
equation of an ellipse with a = b = r. But then we should have the relation
b^{2} = a^{2}(1e^{2}), which implies that e = 0. We can think of this as the limiting
position of the ellipse x^{2}/r^{2} + y^{2}/r^{2}(1e^{2}) = 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 : x^{2} + y^{2} = 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.

