Final answer:
The Fifth Amendment entails that the government can use eminent domain to take the land needed for the transit station but must compensate the homeowners fairly (Option 3). It does not mandate double payment, alternatives studies by the government, nor does it give the state power over residents property without compensation.
Step-by-step explanation:
The Fifth Amendment affects the case of a local government planning to build a new transit station on a site where several homes are currently located by stipulating that the government can take the land through the power of eminent domain but must provide just compensation to the homeowners. Therefore, Option 3 is correct: It says that the government can take the land but must pay a fair price to the owners. This clause is known as the Takings Clause, and it requires the taking to be for public use.
The term 'public use' has been interpreted by the courts to include public benefit or public purpose, as was seen in the case of Kelo v. City of New London, which concluded that economic development projects can qualify as public use. However, the Fifth Amendment does not require the state to pay double the property value, nor does it state that the government must first do a study on alternative sites, though in practice such studies are often undertaken as part of the planning process.