The locations of Beta Geminorum and Pollux on "The Explored Galaxy" star chart
The Beta Geminorum system (or the Pollux system) was a system. Its primary Beta Geminorum was orbited by at least five planets. The closest starbase was Starbase 12. (TOS: "Who Mourns for Adonais?")
When surveying Pollux V and the rest of the Pollux system in 2267, the USS Enterprise detected a statistically unexpected absence of intelligent life at first. However, the ships's crew soon encountered Apollo, the last member of his species of ancient intelligent humanoids, on Pollux IV. (TOS: "Who Mourns for Adonais?")
In 2293, the location of Beta Gemiinorum in the Milky Way Galaxy was labeled in a star chart that was in Captain James T. Kirk's quarters aboard the USS Enterprise-A. (Star Trek VI: The Undiscovered Country, production art)
In 2371, the location of Beta Geminorum was labeled on a star chart Data and Picard were studying in stellar cartography on the USS Enterprise-D. (Star Trek Generations, display graphic)
Background information
Michael Okuda, who created the star chart The Explored Galaxy, depicted Beta Geminorum and Pollux as two distinct stars, even though they were established to be synonymous in "Who Mourns for Adonais?".
According to Star Trek: Star Charts (pgs. 60, 62), the Beta Geminorum system was located in the Beta Quadrant. Beta Geminorum was a K-class star with a magnitude of +3, which was ten times brighter than Sol. In the mid-22nd century, this system was located in non-aligned space. In the late 24th century, this system was located in or near Federation space.
Apocrypha
Pollux V was also featured in the video game Star Trek: 25th Anniversary.