Final answer:
Betelgeuse is the odd star out in the list because it is a red supergiant, which is distinct from Vega, Polaris, and Alpha Draconis (Thuban), which are characterized as being pole stars at different times due to precession of the equinoxes. Betelgeuse is also the coolest star compared to Archenar (blue) and Capella (yellow), as red stars are cooler than blue and yellow stars.
Step-by-step explanation:
The student's question is about identifying the odd one out among a list of stars based on a specific characteristic. To answer the question, we first need to understand that stars can be categorized by their spectral types, which relate to their surface temperatures, with blue stars being the hottest and red stars being the coldest. Looking at the list: Vega (a blue star), Betelgeuse (a red supergiant), Polaris (a bright, yellowish star, commonly known as the North Star), and Alpha Draconis (Thuban, also formerly the pole star), it's evident that Betelgeuse is different in terms of being a red supergiant, whereas the others have been or currently are pole stars due to the precession of the equinoxes. Therefore, the odd one out in this list is Betelgeuse.
When we explore the stars' temperatures and colors further, as indicated by the question of which star is the hottest and which is the coolest among Archenar (blue), Betelgeuse (red), and Capella (yellow), we can conclude that Archenar is the hottest because it's blue, and Betelgeuse is the coolest because it's red. The color of a star reflects its temperature, with blue stars being significantly hotter than yellow and red stars. Betelgeuse is also renowned for its size, large enough that if it were placed at the center of our solar system, its atmosphere would extend past the orbit of Jupiter. This makes Betelgeuse distinct from the rest of the stars listed in several aspects.