Are you using Clark's 66, Mead, WGS 83 ...
There are too many different ellipsoid definitions to attempt to describe a specific algorithm.
If given a longitude and latitude, you have the intersection.
Line 1: Starts at (Lat1s,Lon1s) and ends at (Lat1e,Lon1e)
Line 2: Starts at (Lat2s,Lon2s) and ends at (Lat2e,Lon2e)
If that is the case, I have found the easiest way is to convert the geographic coordinates to Lambert conformal or mercator value and then solve.