Prepositions with Disgusted

The idiomatic prepositions are by, at and with. By is used when disgusted forms part of a passive voice or when it is fully participial in force.

• We were disgusted by the sight. (The passive version of 'The sight disgusted us.')
• Disgusted by what we had seen, we hastily withdrew from the crowd.

"With" is used when the disgust is excited by a person, or by some inanimate thing which is thought of as behaving as a person might:

• I am disgusted with you. 
• I am disgusted with the weather.

"At" is used of some kind of activity or occurrence:

• I am disgusted at your behaving in this way. 

Some nouns, such as conduct, behavior, offer, can take either "with" or "at," according to the idea we wish to express.

'I am disgusted with his conduct' refers to the kind of conduct; 'I am disgusted at his conduct' means 'at his conducting himself in such a way'. 'They were disgusted with the offer' refers to the terms of the offer; 'They were disgusted at the offer' means 'at the fact of such an offer being made'.

"Disgusted" may also be followed by an infinitive or a noun clause: 'I was disgusted to hear of his behavior'. 'I am disgusted that he should behave in this way.'

Reference: "English Prepositional Idioms" by Fredrick T. Wood.

Prepositions with Farm

A person farms land, farms so many acres, farms in a certain locality (e.g. in Dorset, in Kent, in Wiltshire). A farm-worker, or anyone engaged either permanently or temporarily in agricultural work, works "on" a farm (‘During the university vacation, he worked on a farm’), but someone who is doing work not connected with farming but on the farm premises, works "at" a farm (e.g a person doing domestic work, a builder building or repairing outhouses). Even for farm-workers, ‘at’ is used if a particular farm is specified: ‘He works at Blackmoor Farm.’ ‘That is the farm at which I used to work.’ (Cf. work in a factory, but work at a particular factory that is named. )

Reference: English Prepositional Idioms (book)