Check if a latitude and longitude is within a circle

What you basically need, is the distance between two points on the map:

float[] results = new float[1];
Location.distanceBetween(centerLatitude, centerLongitude, testLatitude, testLongitude, results);
float distanceInMeters = results[0];
boolean isWithin10km = distanceInMeters < 10000;

If you have already Location objects:

Location center;
Location test;
float distanceInMeters = center.distanceTo(test);
boolean isWithin10km = distanceInMeters < 10000;

Here is the interesting part of the API used:
https://developer.android.com/reference/android/location/Location.html