What does it actually mean for two bodies to "orbit their center of mass". Does that mean that the two bodies move in ellipses, and the center of mass is a foci of each ellipse?
It is also called a barycenter. Any two (or more) objects in orbit around each other all orbit the barycenter. When working with 2 objects, the center of mass is the barycenter. I think you are confusing the center of mass for each object with the center of mass for the system.
Suppose we are given two masses with non-collinear initial velocity vectors. Assuming the only relevant force is the gravitational attraction between the two masses, does it follow that the bodies orbit their center of mass? In other words, under what conditions do two bodies orbit their center of mass?
The 2 object must always orbit their center of mass of the system.
As for solving the question, it appears there is not enough information provided to determine the period.
Added (corrected):
The periodperiod can be found from:
$T=2\pi \sqrt {\frac{r^3}{GM}}$$T=2\pi \sqrt {\frac{r^3}{G(M_1+M_2)}}$
The more masive body ismasses are M and 3M and the radiusradii of the orbitorbits is $\frac{1}{4}d$$\frac{1}{4}d + \frac{3}{4}d$.
Substituting the values gives:
$T=2\pi \sqrt {\frac{(\frac{1}{4}d)^3}{3GM}}$
$T=\frac {\pi}{4} \sqrt {\frac{d^3}{3GM}}$
Which is not one of the answers. Either I made a mistake, or there is not a correct answer.
If I use $\frac{3}{4}d$ instead, I get:
$T=2\pi \sqrt {\frac{(\frac{3}{4}d)^3}{3GM}}$$T=2\pi \sqrt {\frac{d^3}{4GM}}$
$T=\frac {3\pi}{4} \sqrt {\frac{d^3}{GM}}$$T=\pi \sqrt {\frac{d^3}{GM}}$
Which is BA.