The correct answer is Bettiah.
Vidyapati was a Maithili poet who lived in the 14th and 15th centuries. He is considered one of the greatest poets in the Maithili language. His works include poems, songs, and plays. He is best known for his love poems, which are often set in the context of the Ramayana and Mahabharata.
Bettiah was a princely state in Bihar, India. It was ruled by the Maharajas of Bettiah. The Maharajas were patrons of the arts and literature, and Vidyapati was a favorite of the court. He is said to have written many of his poems for the Maharajas.
Rajnagar, Gidhaur, and Dumraon were also princely states in Bihar. However, there is no evidence that Vidyapati was associated with any of these states.
In conclusion, the correct answer is Bettiah.