That color around the crack means moisture. Maybe from a leak, maybe from being careless with kids in the tub, etc.
What I would do, personally, is cut away the dry wall in that area, from one "beam" to the other in the ceiling, the entire length of the crack. As long as your area being cut out is less then one sheet of drywall, your not spending any "extra" money. If it's larger then a sheet of drywall then you will need to adjust, but it shouldn't be.
The reason I say cut out such a huge piece is simple. Proper repair will have you do that anyway. A patch job that is more then just Spackle, means first cutting a easy to fill shape. From beam to beam, is damn easy, you don't have to do any weird stuff to hold the patch in place. Next is the water. With that large of a section removed you can go turn on every faucet in the house, one at a time, for at least 10 mins. Then check to see if the "beams" get wet. It's really important to run "every" faucet because water can go an insanely long way. Water on one side of the house could be clear from the other side. There is no reason to run water on the first floor. Water can "go up" wood, but it wouldn't look like that and you would see the colors along walls.
With the big hole, you should be able to get a general idea where the leak is coming from. If you don't see any water, and you don't think there is a lean then apply a generous layer of flour (for cooking) to all the "beams" in that area. The flower will get wet if there is even a little water.
If you have a leak call a plumber, they will appreciate the big hole. If you don't have a leak, then check the caulking around the tub. You should be able to pour a bucket of water on the floor in the bathroom and it should stay on the floor, not soak into it. Don't pour a bucket. But with the flour in place maybe pour a glass. If you see water then, you probably need to re-caulk your tub. This is an easy one if you have kids. They splash water, and it goes under the tub into the floor and "leaks" to the floor below.
If you don't see water then, then just move on with the repairs. If you do see water, then repair the leak(s), and move on with the repairs.
To repair the now gaping flour filled mess, you shop vac up the extra flour. If it didn't get wet it should be really easy. Then you give all the wood inside the area a really good spray with bleach (if there was water this will really help avoid mold). Put the new drywall in place and secure it (screws work best, but nails are ok) Then finish it like normal. Drywall tape, paint and blend.