The correct answer is option 2: Dwarasamudra. The capital of the Hoysalas was Dwarasamudra, also known as Halebidu.

Option 1, Malkhed, is incorrect. Malkhed was the capital of the Rashtrakutas, another dynasty that ruled in the Deccan region of India.

Option 3, Somnathpur, is incorrect. Somnathpur is a small town in Karnataka, known for its magnificent Hoysala temple, but it was not the capital of the Hoysalas.

Option 4, Badami, is also incorrect. Badami was the capital of the Chalukyas, an earlier dynasty that ruled in the Deccan region.

Therefore, the correct answer is option 2: Dwarasamudra. It is worth noting that Dwarasamudra was the capital during the reign of the Hoysala king Vishnuvardhana, who played a crucial role in the expansion and development of the Hoysala kingdom. The city is renowned for its exquisite Hoysala temples, including the famous Hoysaleswara Temple.