Your question identifies that the cat will "urinate in one specific area of the carpet". This is a an issue that we see sometimes in rabbits, they decide "this is the place!"
The solution with rabbits is to put the litter box where "they want it to be". Occasionally people are able to slowly (over days or weeks) to move the litter box to where they person prefers it to be, and the pet adapts. But first put the litter box where they want it, and leave it there for a few days. It is also possible that the litter box will need to stay where "they want it to be".
If the area is a bit bigger than the litter box you can use the solution heresolution here to cover a wider area.
