Best way to fix drywall ceiling
Take the left side over to the next joist, add a piece of 2*4 that runs the length of the cut to the side of that joist. This will be your nailer for the new piece of board.
On the right side, cut a chunk of 2*4 again, but this time lay it flat and screw in from the light side of that joist. This will give you something to screw the new and existing board to, much like what Joe stated.
You could also add a flat 2*4 on each end that would span between each joist if you like. Probably wouldn't be a bad idea so that way all seems are secured properly.