Common transversals and tangents to two lines and two quadrics in P³

We solve the following geometric problem, which arises in several three-dimensional applications in computational geometry: For which arrangements of two lines and two spheres in R^3 are there infinitely many lines simultaneously transversal to the two lines and tangent to the two spheres? We also treat a generalization of this problem to projective quadrics: Replacing the spheres in R^3 by quadrics in projective space P^3, and fixing the lines and one general quadric, we give the following complete geometric description of the set of (second) quadrics for which the 2 lines and 2 quadrics have infinitely many transversals and tangents: In the nine-dimensional projective space P^9 of quadrics, this is a curve of degree 24 consisting of 12 plane conics, a remarkably reducible variety.