Rational solutions of certain equations involving norms

Let k be an algebraic closure of k. In the case when P(t) has at most one root in k, the open subset of the affine variety (1) given by P(t)y~O is a principal homogeneous space under an algebraic k-torus. In this case it is well known that the Brauer Manin obstruction is the only obstruction to the Hasse principle and weak approximation on any smooth projective model of this variety (Colliot-Th~l~ne and Sansuc [CSanl]). In this paper we prove the same result when P(t) has exactly two roots in k and no other roots in k, and k is the field of rational numbers Q. An immediate change of variables then reduces (1) to the equation ta~