batter swings, ball hits homeplate then stays fair.
fair or foul

Fair. But even if the ball hits <i>behind</i> the plate, if it then goes fair, it's fair. If the ball settles on the plate, it's fair. The plate is in fair territory.

The only thing special about the home plate and the ball is regarding a pitch.

If a pitch hits home plate, it is by definition a BALL (unless the batter swings). (fast pitch)

Otherwise, home plate is no different from any other part of fair territory. Where the ball is first touched or comes to rest defines its fair/foul status, not where it first strikes the ground (talking inside of 3rd / 1st here).

Oh - I forgot to directly answer your question - FAIR, as greymule said.

thanks for your replys I forgot to mention that this is fastpitch/ any difference?

No difference, coach.

thanks