The centres of a ring of mass m and a sphere of mass M of equal radius R, are at a distance $\sqrt{8}R$ apart as shown in Fig. The force of attraction between the ring and the sphere is 

Refer to figure Let µ be the mass per unit length of the ring. L = 2$\mathrm{\pi }$R  is the length of the ring. Consider a small element of length dx of the ring located at C. Then 

Force along BC is $f=\frac{G M \mu dx}{{\left(3R\right)}^2}.$ Therefore, force along BA is dF = f cos $\theta$ = $\frac{G M \mu dx}{9 R^2}\frac{\sqrt{8}R}{3R}$

                                                                                                                         =$\frac{\sqrt{8}}{27}\frac{G M \mu dx}{R^2}$

$\therefore$ Total force =$\frac{\sqrt{8}}{27}\frac{GM}{R^2}{\int }_{ }\mu dx=\frac{\sqrt{8}}{27}\frac{GMm}{R^2}$

 because ${\int }_{ }\mu dx=\mu \times L=m$, the mass of the ring. Hence the correct choice is (a).