If the leak is occuring right around the tub drain fitting probably the plumbers putty seal under the tub basket has dried out and cracked.
You should be able to complete the repair from inside the tub without tearing the ceiling out, providing you can get the tub basket out.
Use a basket wrench to pull the tub basket out, apply a small rope of plumbers putty about the diameter of a lead pencil on the underside of the basket flange and firmly tighten the basket back into place. You will see the excess putty squeeze out as you tighten it.
You can buy an inexpensive tub basket wrench at any local hardware or big box home supply center for about $5.
