I recently noticed something unusual about the ceiling in one of rooms
in my house (actually, my parents house) and it has me a little
concerned. I'm not sure how to describe it though. There are two vents
in the ceiling, one is to the south end of the room and the other at
the east end. If I lay on the floor and look up at the ceiling, I can
see what looks like marks or imprints of some sort of where the metal
ducts in the attic go from the entrance of the room (west side) over to
the east vent. Basically, these marks go almost all the way across the
room.

There are other weird marks that go from south to north but I have no
idea what they are. There are about 3-5 of them and they're a sort of
light brownish color that are about 4-6 inches wide. My first
impression upon seeing them is that something in the attic got too hot
and it burnt the ceiling. The other thing I thought it could be was
some sort of water damage but I'm not sure that's it.

When I go up to the attic to look, I don't see anything unusual but I'm
not an expert and I'm not sure what to look for. There's nothing in the
attic except for the attic fan, cable tv wires, antenna wires, metal
air ducts, electrical wiring, and insulation. It's not used for storage
as it wasn't designed for that.

The only thing I know is that once, two months ago, there was a small
amount of water dripping from the corner of the east vent in that room.
My father checked it out and it hasn't leaked since then but I don't
know what the problem was. Sometimes, when it rains, it does sound like
there's water dripping in the attic but I can't confirm that.

Any ideas on what could be going on with the marks from the air duct
and the brown marks?