Thirumalai Nayak is known for the construction of the Thirumalai Nayak Palace in:

The correct answer is: a) Madurai.

Thirumalai Nayak was a Nayak ruler of Madurai from 1623 to 1659. He was a great patron of art and architecture, and is best known for the construction of the Thirumalai Nayak Palace in Madurai. The palace is a UNESCO World Heritage Site and is one of the most important examples of Nayak architecture.

The other options are incorrect because:

• Thanjavur was the capital of the Nayak kingdom of Thanjavur, and was ruled by a different Nayak dynasty.
• Vellore was a fort town that was under the control of the Mughal Empire.
• Gingee was a fort town that was under the control of the Vijayanagara Empire.