How can I patch the polyurethane on pine floors?

I have to do a polyurethane repair on the oak floor in my kitchen as well. I don't want to have to sand the entire kitchen floor in order repair a 6 sq ft area. I'm going to use my belt sander to sand down to the bare floor in the area I'm going to repair. Also, I want to "feather" the repair so I'll leave the boundary transition region from bare wood to old poly finish somewhat broad. Then I'll poly the bare wood and try to blend the repair to the old finish, hoping that the new poly and the old aren't too dissimilar. As far as the tack cloth is concerned, I remember reading an article in a woodworking magazine, As I recall, you soak the cloth in a mixture of turpentine and varnish. Ring it out and then keep it stored in a jar when not in use.

Are you going to put a stain (MInwax?) on bare wood to try to match it to the oa color. Was floor originally stained before poly applied?

And does the belt sander have a bag attached to pick up dust? If not, how will you handle that? 6 sq. ft. will generate a lot of it.

I took a furniture refinishing class a while ago and teacher said to brush poly in one direction only and "feather," as you said. No brushing back and forth as one can do w. paint. And don't overload brush. Do two coats, pref. three, instead. Let me know results.