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 ?

Correct — D, Any fixed direction. On a sphere the shortest route between two points is an arc of a great circle, and every great circle on a ball of radius x has the same circumference, 2πx. Setting off from P and holding a fixed direction on a smooth surface means going straight ahead, and a straight-ahead course on a sphere traces out a great circle. The distance travelled, πx, is exactly half of 2πx — so you have gone half way round and are standing at the point diametrically opposite P, its antipode. That single fact settles the question, because through a pair of antipodal points there passes not one great circle but infinitely many, and each of them is a full circle whose half-arc back to P again measures πx. Whichever fixed direction is chosen from the antipode, the journey home is πx and no choice can be shorter. Taking P on the equator gives the same picture concretely: the equator is itself a great circle, half of it is πx, and the traveller ends up on the opposite meridian at the same latitude, that is, at the antipode.

• (a)East only — Carrying on east does bring you home, in another πx, so the direction is not wrong — the word only is. From an antipodal point every direction is equally good, so no single one can be singled out as necessary.
• (b)West only — Retracing the way you came also costs πx, so this is a correct route stated as though it were the sole one. The same objection applies as to the east-only option: the exclusion is what makes it false.
• (c)East or West but not any other direction — The most tempting wrong answer, because it recognises that both ways along the original line work and then stops there. It misses that an antipodal pair is joined by an infinite family of great circles, not by one line with two ends — every one of them is a shortest path, so no other direction is excluded.

A great circle is the intersection of a sphere with a plane through its centre; it is the largest circle that can be drawn on the surface, and the shortest surface path between any two points lies along the one that joins them. For most pairs of points that great circle is unique, which is why long-haul flights have a single optimal track. The exception is a pair of antipodal points, where the joining plane can be rotated freely about the axis through the two of them, producing an unlimited family of equally short routes.

Two things in the stem are doing quiet work. The phrase about a smooth surface so that one can travel along any direction is the licence to treat the journey as a straight-ahead great-circle course rather than as a compass heading held along a line of latitude — a course held due east away from the equator would follow a parallel, which is not a great circle and not a shortest path. And the distance πx is chosen so that the traveller lands exactly on the antipode; any other distance would leave a unique shortest way home and the answer would have been a single direction. Recognising that πx is half of 2πx is the whole insight, and the rest follows without computation.

• The shortest path between two points on a sphere lies along the great circle joining them.
• Every great circle on a sphere of radius x has circumference 2πx, so half of one measures πx.
• A journey of πx along a great circle from any point ends at that point's antipode.
• Infinitely many great circles pass through a pair of antipodal points, and every one of them gives the same shortest distance πx.
• The equator is a great circle; the other parallels of latitude are not, which is why a due-east course off the equator is not a shortest path.

• Assuming the return journey must retrace the outward one, which rules out every other direction for no reason.
• Treating a due-east compass course off the equator as a shortest path; it follows a parallel, not a great circle.
• Missing that πx is half of 2πx and so failing to notice that the traveller is at the antipode.

Asked as a reasoning item dressed as geometry — no calculation is needed once the traveller is recognised to be standing at the antipodal point.

No directly related past PYQ was found.

• practice — not a real PYQ

  Which one of the following is a great circle on the Earth?

  • (a)The Tropic of Cancer
  • (b)The Arctic Circle
  • (c)The Equator
  • (d)The parallel of 45 degrees North

  Answer(c) The Equator — it is the only parallel of latitude whose plane passes through the centre of the Earth, which is what defines a great circle.

• practice — not a real PYQ

  A traveller on a spherical planet of radius R walks straight ahead for a distance of 2πR and stops. Compared with the starting point, the traveller is now

  • (a)at the antipodal point
  • (b)back at the starting point
  • (c)a quarter of the way round
  • (d)at the nearest pole

  Answer(a) back at the starting point — 2πR is a full circumference of a great circle, so the traveller has completed one whole loop.