I am new to ArcGIS and want to implement the Zoom to Selected Features functionality available in Selection menu option (Selection-> Zoom to Selected Features), using C# API.
Is any API available in C# ?
To implement, I would suggest creating an Add-In (in C#) using ArcObjects API. Below are some helpful links to get you started:
At the minimum you will need to install Microsoft Visual Studio 2010 Express and ArcObjects .NET 10.1 SDK.
An alternative would be to develop a python Add-In. This option would entail less code, however you may be more limited on available GIS classes/objects.
As I stated in my comment, a web search should give you some source code to start off with.
Try this one: