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Difference between On someone’s case and On the case

on someone’s case—

1. investigating smb.’s crime:

  • He has been able to avoid getting caught by every other detective on his case.

2. (U.S. sl.) harassing or criticizing a person in an annoying way:

  • “My sister was on my case about that joke all night,” he continued.

on the case

1. = on someone’s case 1:

  • The detectives on the case think the two things are tied in. The killing and the acid.

2. dealing with a particular problem or situation:

  • The missions seem designed only as a chance for politicians to show they’re on the case.