Configurations of 2n-2 quadrics in ℝn with 3 · 2n-1 common tangent lines

We construct 2n - 2 smooth quadrics in ℝn whose equations have the same degree 2 homogeneous parts such that these quadrics have 3 · 2n-1 isolated common real tangent lines. Special cases of the construction give examples of 2n - 2 spheres with affinely dependent centres such that all but one of the radii are equal, and of 2n - 2 quadrics which are translated images of each other.