The client set up is that to consume SP 2013 search service from 2010 search center. The connection has been set up, but a new requirement has a need to find out the Global search center url of the 2013 search application from the 2010 search result page. Basically by clicking a button in 2010 search result page, user will be re-directed to 2013 search page (which exists but serves a whole different purpose from 2010 search page). The 2013 search center url might be different based on the environment it is being accessed from. Is there a way this can be achieved using client side javascript?

Here's my thought process so far, but not sure if this is achievable. 

1. From the 2010 search result page, identify the search service application that is being consumed. (This is cross farm and not sure how to handle this).

2. From the search service application, get the global search center url.

Hi,

According to your description, my understanding is that you want to get SharePoint 2013 global search center url in SharePoint 2010.

In this case, as it is cross farm and there is no Client API to get the search center url directly, I suggest you can create a custom web service with C# code in SharePoint 2013 and then you can call the web service using Jquery.

Here are some detailed articles for your reference:

Get the Search Service Application for a specific site (programmatically)

Walkthrough: Creating a Custom ASP.NET (ASMX) Web Service in SharePoint 2010

Using jQuery with custom Web services in SharePoint

Thanks

Best Regards