The mother ask the courts to ask me and I did do I have to pay child support

If ordered, yes, you will still have to pay support.

Typically "giving up your rights" means that you will no longer be make legal decisions regarding the health and welfare of your child. You will still be liable for support up until the child is adopted by a step-parent.

Are you sure your rights were terminated? It is rare for a court to actually terminate parental rights. And they will almost never do so if it means stopping a support obligation.

What probably happened is you were given no legal or physical right to see the child, but you still have a support obligation if the court orders it.

You need to read the court orders.

Unless there was an adoption, or you are a serious threat to child, rights are not taken away. Often the parent may lose custody, may have lost rights to visit, and others.

But you have to pay, as long as there is court order