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That is an awesome question!
They can tell if a star is part of a binary system the same way they could tell that there were supermassive black holes in the middle of galaxies, that solar systems orbit stars, and that moons orbit planets. By looking at the way that the bodies interact.
You can tell all of these things several ways, but the easiest is by using the big fancy Keplar telescope we've setup in space to look at the minute differences in light when one of them passes in front of the other, if that continues to happen, but with both of those stars constantly passing over each other, and while thats happening other bodies are then passing around those stars. It becomes obvious its a binary system!
However, an alternative way, when they do not have a solar system around them, is that 2 stars, very close together, are orbiting very closely, but not quite around each other, instead circling somewhere in the middle.
Binary systems have their own orbits, but they are affected by each others gravity as well, creating this central point in which both orbit.
Cool thing, these orbits don't even have to be the same size!
Now! Ontop of that, this is just with 2 stars, if you're meaning a black hole and a quasar, it becomes a whole different ball game! The orbits are still similar, however the biggest indicator is the way that the black hole essentially "Feeds" off of the star!
I hope this helps! ;)
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