Twenty-five people were charged Thursday with obtaining at least $75 million in fraudulent Medicaid payments from the District of Columbia government, a series of cases that federal prosecutors said added up to the largest health-care fraud in the city’s history. U.S. Attorney Ronald C. Machen Jr. said agents fanned out across the D.C. region to make arrests and seize six luxury vehicles and 49 bank accounts, capping off a multiyear investigation. Prosecutors said personal health care in private homes is a thriving front in Medicaid fraud, a category of Medicaid claims that has grown dramatically in the city over the past 8 years, and said there have been similar cases out of Miami and Detroit. The uptick in billings for home care — from $40 million in 2006 to $280 million last year — was part of what tipped off authorities to illegal activity, U.S. Attorney Ronald Machen said. Prosecutors say many of the defendants persuaded patients to fake illness or injury so they could bill Medicaid for home care they didn’t receive. Some of those patients received kickbacks, authorities said, although no patients have been charged. Those charged include the owners of home-care firms and nurse staffing agencies, home-care aides and recruiters who signed up patients.