Hi

There's no compulsion of reporting at the same participating Institute (in your case NIT Trichy). You can report anywhere but the institute should be mentioned in the list of all the colleges in JoSAA 2019. To avoid delay and hotchpotch please report at any of the nearest reporting centres.

For more information, do visit the JoSAA 2019 Official Site. Also you can always check the same about JoSAA Counselling 2019 through this link: 

Best of Luck. 

Hope this helps.

Cheers!