Thin blue line on the screen of my iMac

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.

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

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