Why are you so "bent on this" is the correct expression. = determined to do something. https://dictionary.cambridge.org/dictionary/english/be-bent-on-sth-doing-sth "why are you so bent on starting an argument" "why are you so bent on getting your own way" "why are you so bent on leaving me at home and going to the pub with your mates" "why are you so bent on not listening to a word I say" "why are you so bent on being such a pain in the !!" "why are you so bent on making trouble" And so on = why are you so determined. "The other bents" are not related it is an expression/idiom

It would, I'd say, but I've never heard "bent" alone - the expression is normally "bent out of shape". "Bent" alone can mean corrupt and dishonest - "a bent politician". It also is a slang term (maybe used more in the 1950s-1970s) meaning strange or unusual (and usually transgressive) behavior. The connection between them all is "out of the ordinary" - think of a metal rod that has been "bent" and twisted into a different shape.

I agree with John. "To be bent ON something" means that you insist on doing something despite people wanting you to stop. It may imply that you're obsessed with doing something, but not crazy. There's another saying that uses the word bent. It's a little dated, but it's "to be bent OUT OF SHAPE BECAUSE something" and it means something upset or saddened you. For example, "he's bent out of shape because he didn't get the job"

angry / upset / uneasy / bothered