Yes you are eligible to apply but sorry chances of you getting a seat in Karnataka government or even private medical colleges stand nil. Since last year the last person to enter Karnataka government medical college under 15% AIQ cutoff had 9154 rank with 542 marks. 

Chances of you under 85% state quota with the rank you mentioned are almost rare as well. 

So the better option left with you is either to take a drop and comeback next year with more stronger and better marks or else go for other courses if you feel interested like BAMS BHMS or BPT or BSC or even pharmacy or something like that.

To get a detailed information about the cutoffs of Karnataka refer to our site...

https://medicine.careers360.com/articles/neet-cutoff-karnataka

Best wishes. 

Thank-you.