\nThe Earth is a sphere of radius x km. A great circle on this sphere has a circumference of 2\u03c0x km. Travelling a distance of \u03c0x km is travelling exactly half the circumference of a great circle.
\nThe question states you travel “along the East direction a distance of \u03c0x km”. Travelling purely East means moving along a circle of latitude. A circle of latitude is a great circle only if it is the Equator (latitude 0).
\nIf P is on the Equator, travelling East a distance \u03c0x along the Equator will take you to the point antipodal (exactly opposite) to P. Let this point be Q.
\nFrom Q (the antipodal point), the shortest distance back to P is along a great circle, and this distance is \u03c0x km. You can travel along the Equator back to P, either East or West. Both directions along the Equator lead to P in a distance of \u03c0x.
\nOther great circle paths from Q to P also exist, for instance, travelling North along a meridian to the North Pole (distance \u03c0x\/2) and then South along a meridian from the North Pole to P (distance \u03c0x\/2), totaling \u03c0x. Similarly, going via the South Pole takes \u03c0x. The initial directions from Q along these meridian paths are North and South respectively.
\nHowever, option C specifically limits the directions to “East or West but not any other direction”. This makes sense only in the simplified scenario where P is on the Equator, Q is antipodal, and the considered return paths are limited to East or West along the Equator. In this specific (likely intended) case, starting East or West from Q along the equator constitutes a minimum distance path back to P.
\n<\/section>\n