DbGeography search query
Instead of doing SELECT TOP 10 * FROM ... you should only retrieve the columns needed which in your case is LineName.  
var name = "" doesn't need to be there, if you just return firstOrDefault.LineName and for no result/exception return string.Empty.  
While we are at var firstOrDefault I would like to say that using var here is just wrong because it isn't clear from the right side of the assignment what the type is. The variable name firstOrDefault isn't descriptive enough so you should consider using maybe geographyLine or foundGeographyLine.  
You are selecting the TOP 10 but using FirstOrDefault. This just doesn't seem right. Using SELECT TOP 1 would be much better and faster.