Cut open the ceiling and look.

It is usually far easier and cheaper to fix than a floor.

First get some estimates on the firstborn.
Then do as Brickeyee says. If you remember where the stain first started you should open up there. The best thing is to have "x-ray vision" and imagine the fixture layout from below so you'd have the likeliest source of the drip.
Like a leak anywhere, it could be running along pipe, joist, romex etc., so start with a small hole.
The stain is not indicative of the source.

The COLOR of the stain is not indicative of the source.

Thanks, brickeyee.

There are other factors, which is why I'd like to know if there was a preference from a plumbing perspective.

The ceiling is popcorn in an open floor plan, not conducive to spot repair. It might be better to cover the whole ceiling with 1/4" to get rid of the popcorn, rather than just a repair. Then I could put in can lights at the same time ... another plus, but also another expense. I've done drywall ceilings myself before, but that was 15 years ago. Hiring someone would be necessary this time cause I'm so old. The ceiling isn't damaged right now, just stained a little, so priming and painting would probably fix it. It's about 700 sq. ft. I have no idea what it would cost to have it covered with 1/4" and mudded, but that would probably be the way to go.

The bathroom floor is original sheet vinyl, in need of replacement. Plus, the geniuses who put it in used nails rather than screws to attach the plywood subfloor. The nail heads have poked through in a couple of places. If I pulled up the floor, I could replace the nails with screws, and put down new vinyl or tile. I did my kitchen with TrafficMaster Allure a few years ago and love it. No problems, and it still looks like new several years later. It's about 100 sq. ft. I could fix the floor of the bath myself. That makes three checks in the floor column.

It's a tossup as to which would be the better upgrade. Both are on the "eventually" list.

Hi, alphonse. Thanks.

With a small hole (how small) what to use to see inside?

Firstborn: not worth a nickel, but wouldn't take a million bucks for him.

Xray specs: don't work worth a darn. All I see are feathers.

Best thing is to find the source of the problem, and the ceiling IS the most expedient, cheapest fix.

Is the "three feet long by a couple of inches wide" stain aligned with a joist?

If so determine which side to open up on. Unless you're good with mirrors the access likely need be at least the size of your head.

Likely sources (from what I see here) could be seal broken on the wax ring, corroded tub drain, possible crack in the fibreglass unit (especially if the builder didn't support properly), pinhole piping leaks in copper from acid water..you did say built in '84-85...

From a plumbing standpoint, i.e. hiring a plumber, best to find the leak. You'll pay his time for that, less for a rocker to patch up. If you're lucky and it's a small patch that can be DIY. A bad wax ring is obviously fixable from above but surest proof is below. If the piping or drain needs replacement, I'd rather do that overhead unless ripping out the bathroom IS something you want to do. But now you are talking some bucks.

I have to point out the leak might not even be plumbing related.

Followup: It turned out to be a leak in the bathtub hookup. The plumber was able to find it quickly and fix it.

I couldn't match the ceiling paint (over the popcorn), so I decided to I spend what I'd budgeted for the plumbing on the ceilings. I had the popcorn ceiling covered with 1/2" drywall anyway and had crown molding put up. It looks fabulous.

Before they put up the new drywall, I cut a hole in the ceiling to make sure the leak had been definitively fixed. It was. I took pictures for the future.

Thanks for all the help, Alphonse and all.

You're welcome, and thanks for the feedback. Glad it didn't cost any appendages or the firstborn.