Private engineering colleges in India conduct an entrance exam of their own for admissions. They don't have reservations for any category. 82 is a decent percentile but it is mandatory to write the entrance exams. A few reputed colleges are-

• Birla Institute of Science and Technology, Pilani
• Manipal Institue of Technology, Manipal
• Vellore Institute of Technology, Vellore
• SRM University, Chennai
• IIIT Hyderabad
• Birla Institute of Technology, Mersa

and many more. Kindly use the following link which will help you predict the colleges you can apply to using your JEE Mains percentile.

https://engineering.careers360.com/jee-main-college-predictor?icn=QnA&ici=qna_answer

I hope this helps.