Assume that the Earth is a spherical ball of radius x km with a smooth surface so that one can travel along any direction. If you have travelled from point P on the Earth's surface along the East direction a distance of πx km, which direction do you have to travel to return to P so that the distance required to travel is minimum? (a) East only (b) West only (c) East or West but not any other direction (d) Any fixed direction

The Earth is modeled as a sphere with radius x. The circumference of a great circle on this sphere is 2πx [1][2]. Traveling a distance of πx km along any direction (including East) from point P means you have covered exactly half the circumference of a great circle, reaching the antipodal point of P [2]. Antipodal points are diametrically opposite locations on a sphere. According to spherical geometry, any two antipodal points are connected by an infinite number of great circles [2]. Since a great circle represents the shortest path (geodesic) between two points on a sphere, and all such paths from an antipode back to the original point P have an equal minimum length of πx, you can travel in any fixed direction to return to P with the minimum distance [2].

Sources

1. [1] Certificate Physical and Human Geography , GC Leong (Oxford University press 3rd ed.) > Chapter 2: The Earth's Crust > Great Circle Routes > p. 14
2. [2] https://en.wikipedia.org/wiki/Great-circle_distance