If I wash my hands to remove impurity under the tap which is fixed to a bath, where the water goes through the drain, do I then have to wash the bath because the water touched the bath before going through the drain?

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.

If the water that comes down from removing impurity is not changed by this impurity, then it is ruled to be pure. Hence, if the water that runs down from washing your hands is not changed with impurity, then the bath does not become impure. However, if the water that runs down from your hands has changed, then the bath becomes impure.

Note that the water that runs down afterwards on the place of this impurity is not changed and thus purifies the effect of this impure water, if we presume that it is impure in the first place. In addition to this, purifying the bath is not an obligation in principle if we presume that it has become impure with the water, because it is only an obligation to purify the body and the clothes in order to perform the prayer, as well as the place where the prayer is held.

From your question, we perceive that you are affected with some kind of Waswaas (obsessive whisperings; OCD); so if this is the case, then you should avoid it and not pay any attention to it, because getting carried away with such whisperings leads to great evil.

Allah knows best.