hello,

there are no category as private. If you don't have any reservation then you will go under general category. You will get AIR(common rank list)  and general category rank. You can use it and go for counselling and apply for colleges which provide petroleum engineering as a degree. If your rank qualifies then you will get an admission through counselling.

If your rank is very big and doesn't satisfies then you have to take direct admission in any private institute.

You can get a good university as per your choice if and only if you score good in the state level or national level engineering entrance exam.  JEE main is accepted by mostly all the colleges. every state has its own entrance exam too. so it depends which exam you will appear and what are the option you will be eligible for and your rank. These factors will decide where you will get an admission.

The better you perform in the exam , you will get more choices of good and best top college and universities as option for admission.

Hope it helps!