What type of star system is characterized as a Binary and Cepheid variable star?

The correct answer is a type of star system that is specifically identified with the term "Cepheid variable star," which is exemplified by the star Delta Cephei. Cepheid variables are a type of luminous variable star that exhibit a regular and predictable pattern of brightness changes due to pulsations in their outer layers. Delta Cephei, in particular, is the prototype of these variable stars, serving as a benchmark in the distance measurement of galaxies.

In double systems, especially those involving Cepheid variables, the term binary can apply if two stars are interacting in a manner that influences each other's properties or brightness. Delta Cephei itself has a binary companion, thereby highlighting its classification as both a binary star and a Cepheid variable. This makes it a focal point for studies on stellar evolution and distances in astronomy.

Other options present stars that do not fall within the specific definition of being a Cepheid variable binary star. Alpha Orionis, also known as Betelgeuse, is a red supergiant with different characteristics. Cygnus X-1 is notable as a black hole binary system. V838 Mon is a peculiar variable star but does not fit the characteristics of a Cepheid variable.