The book Nyaya-tattva, which gives an elaborate exposition of the philosophy of the Vishishtadvaita school, was authored by:

Correct option is A

The book Nyaya-tattva, which provides an extensive explanation of the Vishishtadvaita philosophy, was authored by Nathamuni. He is considered the first Sri Vaishnava acharya (teacher). Nathamuni also compiled the Naalayira Divya Prabandham. 

Acharya Lagadha Muni is one of the earliest known Hindu mathematician and astronomer. This great Indian Maharishi, also known as Lagadh Muni, is believed to have lived during 1200 BC. His contribution to astronomy
is still relevant and actual.

Bharata muni: He is traditionally attributed authorship of the influential performing arts treatise Natya Shastra, which covers ancient Indian dance, poetics, dramaturgy, and music.

Ramanuja (1077 – 1157), also known as Ramanujacharya, was an Indian Hindu philosopher, guru and social reformer. He is one of the most important exponents of the Sri Vaishnavism tradition in Hinduism. His philosophical foundations for devotional practice were influential in the Bhakti movement. Ramanuja himself wrote influential texts, such as Sanskrit bhāsyas on the Brahma Sutras and the Bhagavad Gita.