Two questions on three quadrics in $P^5$ whose intersection is a genus $5$ K3 surface.

Question

It is well known that the intersection of three quadrics in $P^5$ yields a genus $5$ K3 surface. (See this link: https://en.wikipedia.org/wiki/K3_surface ).

Question I:

Does anyone have an example (or a link to an example) in which one of the quadrics is a hyperboloid of one sheet and one is a hyperbolic paraboloid?

Question II:

Assuming that such a case can exist, is it possible that the intersection can contain one line which is a ruling on both surfaces (the hyperboloid and the paraboloid)?

Or am I not visualizing the situation correctly in the first place?

Thanks as always for whatever time you can afford to spend considering this matter.

Note: in the comments below, I asked LordShark the following question:

So in this context, "quadric" is the kind of "quadric" which appears in this discussion of Plucker coordinates:

https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates

Answer 1

LordSharks's comment above actually answers the question by explaining why the question was ill-formed in the first place.