Universal torsors and values of quadratic polynomials represented by norms

Let K/k be an extension of number fields, and let P(t) be a quadratic polynomial over k. Let X be the affine variety defined by P(t) = N_{K/k}(z). We study the Hasse principle and weak approximation for X in two cases. For [K:k]=4 and P(t) irreducible over k and split in K, we prove the Hasse principle and weak approximation. For k=Q with arbitrary K, we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one.