Under what condition can I termination of child

What exactly do you mean. Do you mean an abortion?

I was thinking terminate rights, but it is unclear

Okay the child mother gave her mother & father legal custody the child. I am current paying court ordered child support to the mother of the child. I just became aware the have given custody to her parent,do I have to continue to pay the child support to the mother. Is there any way to get a copy of the paper work where the child's gave custody to her parents?

The mother can't just give custody. A court awards custody. You should have been informed that she was giving up custody and you should have been given the opportunity to apply for custody. Assuming you are the legal father, then you may have a say here unless the mother had full legal custody.

So you need to find out what court gave custody and get a transcript of that hearing.

Next, who are you paying support to? If you are paying support directly to the mother, then as soon as you see a court order awarding custody to her parents, you can start paying support to the parents. But It would be better to have your support order modified by the court.

If you are paying to an agency, they might already be forwarding your support to the grandparents. But definitely check with the agency.

The only way you can terminate your support is if a court orders it terminated or the child is adopted.

Are you saying I will still have to pay child support even after the mother gave the child away.

Yes, if you did not go in and file for custody to take the child, you did not give up your rights, Most likely the mother gave guardianship to her parents. You should have been served notice. It is very possible she did not even legally do it, more often they drop kid off at parents, sign some paper that is not filed in court and think they signed over rights

But yes as long as there is a court order for support you have t pay for the child. To who ever is the guardian.

Are you saying I will still have to pay child support even after the mother gave the child away.

Of course you do! Why would you think that changing the custody of the child would affect your OBLIGATION to help support that child?

It might change who you pay support to, it might even change the amount of support you have to pay. But to change the amount you have to go back to court for a modification based on changed circumstances. But until the child becomes an adult you are required to help provide support.