Is a person connected to a life support machine, or is the life support machine connected to them?
The common usage is that the person is connected to the life support. This is because usually the person is what's being discussed.
John is connected to life support.
The person, when it's what is important to the idea being talked about, is the subject, the state of being connected is the verb and the life support is the object.
If the life support machine is important to the discussion, then it comes first:
Our only ventilator is connected to a patient already.