When Elizabeth and David take off at the end of the movie Prometheus in the alien ship are they heading for LV-426?
From an interview with writer Damon Lindelof:
In that final scene, David wants to go to Earth, and Elizabeth wants to go to the alien home world. Where do you think they’re going?
I think they’re going where she wants to go. His fundamental programming has been scrapped. Weyland [the man who built and programmed him] is dead and so now his programming is coming from God knows where. Is he being programmed by Elizabeth, or is it his own internal curiosity now that Weyland isn’t telling him what to do any more?
He’s always been interested in Elizabeth, remember that: He’s watching her dreams when she’s sleeping in much the same way that he watches 'Lawrence of Arabia'. He’s a strange robot that has a curious crush on a human being, and when Weyland is eliminated, I think he is genuinely interested in what she’s interested in. He reaches out partly for survival, but partly out of curiosity, and I think he’s sincere that he’ll take her wherever she wants to go.
LV-426 is not the Engineers' homeworld.
According to the new Alien: Covenant trailer movie, Shaw and David went to the Engineers homeworld, the paradise planet.