Among the following 4 answers, which one is correct for the following question and why? - In which direction does the shortest path (on the globe) from $A$ to $B$ go, if the geographical coordinates of point $A$ are: latitude $70^{\circ}$ north, longitude $10^{\circ}$ west, and the coordinates of point $B$ are: latitude $70^{\circ}$ north, longitude $175^{\circ}$ east.

a) exactly east,

b) exactly west,

c) exactly via the North Pole, d) not in any of the above directions.
Among the following, the answer d) is correct. The path b) is shorter than a) because it follows the arc corresponding to a central angle of $175^{\circ}$ and $185^{\circ}$ on the $70^{\circ}$ north latitude circle $k$ of the same sphere.

![](https://cdn.mathpix.com/cropped/2024_05_02_68324bba59026a2d860bg-1.jpg?height=580&width=715&top_left_y=290&top_left_x=694)

Figure 1

The radius of the mentioned circle $k$ is $r = R \cos 70^{\circ}$, where $R$ is the radius of the Earth (considered as a sphere), so the length of the path b) is

$$
\frac{175}{360} \cdot 2 \pi R \cos 70^{\circ}
$$

The path c) is understood as traveling from $A$ to $E$ along the arc of the meridian at $10^{\circ}$ west longitude, and then from $E$ to $B$ along the arc of the meridian at $175^{\circ}$ east longitude. The meridians are great circles of the sphere, with radius $R$, and the central angles corresponding to the arcs are $90^{\circ} - 70^{\circ} = 20^{\circ}$. The two arcs together make up $\left(20^{\circ} + 20^{\circ}\right) / 360^{\circ} = 1/9$ of the circumference of a great circle, so the length of the path c), and its ratio to the length of the path b) is

$$
\frac{2 \pi R}{9}, \quad \frac{8}{35 \cos 70^{\circ}} < \frac{8}{35 \cdot 0.3} < \frac{8}{10} < 1
$$

Thus, the path c) is shorter than b).

We will also show that there is a path shorter than c) from $A$ to $B$. Such a path e) is the shorter arc (i.e., shorter than a semicircle) of the great circle passing through $A$ and $B$, and the chord corresponding to this arc is the segment $AB$, which is also a chord of $k$. The path c), however, is equal to the path that leads from $A$ to the antipodal point $A'$ of $k$ (longitude $170^{\circ}$ east) via $E$, i.e., along a meridian, a great circle, and the chord $A A'$ corresponding to the great circle arc $A E A'$, which is the diameter of $k$, and thus longer than $AB$ (Figure 2).

![](https://cdn.mathpix.com/cropped/2024_05_02_68324bba59026a2d860bg-1.jpg?height=626&width=520&top_left_y=1687&top_left_x=781)

Among the arcs of equal-radius circles that are not greater than a semicircle, the one with the shorter chord is smaller, so e) is shorter than c).

This justifies the correctness of the answer d).

Note. Our questions referred to the Earth as a sphere, but the conclusions of our considerations are also valid for the actual shape of the Earth, the so-called geoid, because the differences in the lengths of the paths are larger, and the change in the Earth's shape would not alter their signs. Several papers mentioned this. Such an in-depth consideration is instructive, but it was not expected in the brief answer to a test question.