Explanation:

- Option 1: Badami

- Correct Answer ()

- Badami, formerly known as Vatapi, was the capital of the Early Chalukya dynasty. It is located in present-day Karnataka, India. The Chalukyas ruled this region between the 6th and 8th centuries.

- Option 2: Kanchepuram

- Kancheepuram, in Tamil Nadu, was never a Chalukya capital. It has been an important center in Tamil history, associated mainly with the Pallavas.

- Option 3: Pataliputra

- This city, in present-day Bihar, was central to the Maurya and Gupta empires, not the Chalukya dynasty.

- Option 4: Rajgir

- Historically significant during the time of the Magadha empire, Rajgir was associated more with the Maurya dynasty, not the Chalukyas.