Hello Himani,
I understand your dilemma. If you can move out of your city then go out and study. That's the first option.

Or you can do your 12th from NIOS ( NATIONAL INSTITUTE OF OPEN SCHOOLING) it provides long distance education and is a valid board and accepted by very government College in India as it is equivalent to cbse,icse and any other board. It comes under MHRD( MINISTRY OF HUMAN RESOURCE AND DEVELOPMENT) of Indian be govt.
You just have pass the exam with very good marks.

Also, one more thing you can do if you are of cbse board and you don't have arts option then look into for state board, there you might get. So, you can switch.

All you need is a good grasp at your subjects and good marks in 12th.

I hope this helps you. And if you have any doubts feel free to comment it.
All the best !!