So when you connect to an external monitor you don't have the lines like you notice when you look at your laptop screen.
Seems like the problem could be a connection because you don't see the line on the external monitor. If you were able to duplicate the same problem when connected to the external monitor the problem will have to be the graphic chip.
Can you send an image of what your screen looks like. Have you tried moving your screen back and forth to see if the line disappears or any other different occurence.
Is your laptop still under warranty?