I think the best approach is for you to store the results in a cache with the center coordinates as a key, and later query the points within the circle for the new request.
I'm not aware of any cache engines that would allow you to perform spatial queries, so I think you'll have to use a database that allows easy querying and indexing of spatial data. You may use that database for caching your results, or at least store a key to that result in a cache engine somewhere else, and later you can query them with spatial coordinates, asking for all points with a threshold distance to your new request.
There's PostGis for PostgreSQL, which should be quite straightforward since it has full support for latitude/longitude distance computations. Once you have it setup with proper indexes, it should be as easy as:
SELECT * FROM your_cache_table
WHERE ST_Distance_Sphere(the_geom, ST_MakePoint(new_lon, new_lat)) <= 160.934
MySQL has some support for the OpenGis extensions, however it doesn't have support for latitude/longitude distance computations. Maybe you'll need to do some calculations by yourself, maybe the simple cartesian distance works for you. Check the documentation here, and this answer should also help.
I also believe even MySQL 5.6 still has support for spatial indexes only in MyISAM tables, but that shouldn't be an issue since you're using them only for cache.
Managing the cache may be a little more complicated than usual. If you need expiration, you should probably store only keys in the database and set an expire parameter on the cache server. When you hit a database point for which there's no longer a valid key, you clean it from the database. You'll probably need a way to invalidate cache when the primary data changes, removing from both the database and the cache server.