Can a patient eat if there tube inserted in the neck for breathing. patient has been on a ventilator for past week and is unable to breath on his own?

Not really: Feedings can be provided by a tube that goes through the nose or mouth into the stomach or through an iv. Usually people do not eat by mouth when they are ventilator dependant.

Yes, sometimes...: Sometimes a person with a trach tube can eat. To determine whether or not the person can eat without aspirating, they will need to have a test called a modified barium swallow. It consists of eating different foods mixed w/ barium and watching where the food goes via x-ray. The test is done by a radiologist and speech therapist and the person can then eat what is deemed safe by the speech therapis.

Yes: It sounds like you are referring to a tracheostomy. This is done when a patient needs to be on a ventilator for a longer time.. Many patients with such a tube can swallow and therefore eat, especially if they have gotten well enough to have periods of time off the ventilator, but some will require artificial feeding for a time via a feeding tube if they are not able to swallow.