With a score of 510, your rank might be approximately between 38822-44553.  This is an expected rank range: https://medicine.careers360.com/articles/neet-result-marks-vs-rank . Now regarding the Colleges that you may have a shot at, you have a chance in:

• JSS Medical College
• K.S Hegde Medical Academy
• Amrita Institute of Medical Sciences

These are not in Bihar but across India.  A seat in Biihar GMCs is tad difficult with this score. But you might get a seat under Management Quota of a Private Medical College. Having said that, this is based on 2019 data and cut offs change every year. Cut offs change every year depending on various factors such as no.of Candidates appearing for the exam, no.of Candidates qualified, top NEET score and difficulty of the exam. For a complete list of colleges that you may have a shot at, please use our College predictor: https://medicine.careers360.com/neet-college-predictor?icn=QnA&ici=qna_answer

Hello student,

Your predicted all india rank is around 45,000 - 55,000. With a score of 510 marks in neet examination, you have less chances of procuring the seat in the government medical colleges when comparing the results of the past years. Yet ranks and scores vary each year depending on various factors like no of students attending the exam, difficulty level of the question paper and state quotas. However you can use the following predictors to find your rank and colleges:

NEET College Predictor Link

https://medicine.careers360.com/neet-college-predictor?icn=QnA&ici=qna_answer

NEET rank Predictor