Question

why did india sends peace keeping force to srilanka

Answer 1

At the beginning of the Sri Lankan Civil War, the Liberation Tigers of Tamil Eelam (LTTE), which was a militant organisation who wanted a separate state for Tamils in North/ East Sri Lanka, caused havoc in the country. Sri Lankan parties were not able to reach a clear solution on their own and this was becoming evident around 1984. 
On Sri Lankan government's request, the Indian government under Rajiv Gandhi assumed a mediation role. There was a peace agreement/ 13th Amendment/ Rajiv-Jayawardene Accord which was signed, for which there were several reasons:

 India, because of a large number of Tamils in the southern part of the country, and also to avoid foreign intervention (USA, Pakistan, China) in the Sri Lankan matters, had been on friendly terms with Sri Lanka and had also promised armed support, if diplomatic interventions failed. Around 1985, Sri Lanka started rearming itself with support from Pakistan, Israel, SIngapore and South Africa. 
As the situation worsened, India was expected to show support because of the huge Tamil population back home. A large number of refugees were entering Tamil Nadu Government in Tamil Nadu, AIADMK, were on friendly terms with Rajiv Gandhi's Congress. They wanted support from New Delhi to thwart that violence. Not doing anything about these wishes could have cost Mr Gandhi the support of AIADMK,
Initially, diplomatic intervention was offered. It was more or less futile. Soon, humanitarian assistance was offered. They started sending aerial supplies. Big cities were sent food, medicine etc. Everything was done to show Indian support.
Soon, Sri lankan president, Jayewardene, met with Rajiv Gandhi and the agreement was signed. Under this, India had to send a peacekeeping force, IPKF, to Sri Lanka, who were supposed to disarm the rebels.

This turned out in a way which wasn't expected, with the LTTE not agreeing to throw their weapons and also attacking the Indian troops. More than a thousand soldiers lost their lives.