Binary Star system with one star stationary? Can a Binary Star system be possible where in one star is stationary and the other star revolves around it? (Just like a planet revolving a star. i.e planets in the system and the star revolving around their own center of masses to balance the system).
 A: In any system, components revolve around center of mass of the system. In our solar system, Sun has 99.9% mass of the system. So, center of mass of system is inside the Sun (nearly coinciding with center of Sun).
Seeing the massive stars in the existence, yes, it's possible that a Sun type star can truly revolve around R136a1 type stars. R136a1 stars are 265 times solar mass. In a binary system with Sun, it'd contain 99.6% mass of the system. So, center of mass of the system would exist near its own center (so, size of star doesn't matter here).
A: Well, motion is relative so you can choose a frame of reference where one is stationary.  If you do though, it makes the equations of motion quite complicated.
Even in our solar system, the Sun isn't stationary.  It orbits the center of mass of the whole solar system (barycenter), just as each planet orbits the center of mass.
The center of mass of our solar system moves relative to the sun due to the motion of the Sun and planets.  Here is a graph:

(Source)
So you could potentially call a body in a system where the barycenter stays inside that body "stationary" but that's not technically correct, no matter how lopsided the masses of the bodies are.  For example:

The larger mass will still always move relative to the system barycenter.
A: As others explained: When two masses interact gravitationally, it's not like the smaller mass is orbiting the larger mass. Both bodies orbit the common barycenter. When one of the two masses is extremely large compared to the other, the barycenter of the system is almost in the center of the larger mass, so the effect on the larger mass is negligible (like a satellite orbiting the earth). But it still won't be completely stationary. The effect would just be too small to measure.
But I would like to get back to your original question, "Can a Binary Star system be possible where the mass difference is so large that the effect on the larger star is negligible": There isn't that much difference in the masses of stars. Stars need to be massive enough to generate enough pressure to maintain nuclear fusion, but not so massive that they collapse into black holes. Also, the larger the star, the faster does it undergo fusion and the shorter its lifespan. For that reason the heaviest known stars have just about 100 times the mass of our sun, while the lightest know stars have just one tenth of a solar mass. A mass difference of factor 1000 isn't small enough that the effect on the larger mass wouldn't be notable.
