An irreducible quadratic polynomial P(X, Y) in two variables over a field k is called a conic over k. It is called rational if its function field is simple transcendental over k (equivalently if P is parameterisable by rational functions). Ohm's rationality criterion states that P is rational if and only if (i) the locus of P is non-empty and (ii) k is algebraically closed in the function field of P. To show the irredundancy of (ii), Ohm gives an example of a non-rational conic with a non-empty locus. That locus, however, consists of a single point.

In this note, we show that a better example cannot exist by showing that if the locus of a conic contains more than one point then it is rational. We also show that the only rational conic whose locus consists of one point is the conic XY + 1 over the field of two elements.