Thin blue line on the screen of my iMac

Make an appointment at your local Apple Store or Apple Authorized Service Provider and let them have a look.

Take a screen shot & view that in Preview, move the Preview window around, if you see double lines it's a Video card/VRAM problem, if the lines do not move moving the pic around then it's the LCD or cable.

Then it's the screen, den.thed has the advice on that.