A solid sphere of mass M and a ring of mass m have their centres lying on the x-axis separated by a distance $2\sqrt{2}\mathrm{r}$ where r is the radius of the ring as shown in Fig. The gravitational force exerted by the sphere on the ring is

The entire mass of the sphere can be assumed to be concentrated at its centre of mass O. Consider a small element of the ring at A. Let dm be the mass of the element (see Fig) 

The gravitational force exerted by M on dm is

$\mathrm{dF}=\frac{\mathrm{GMdm}}{{\left(3\mathrm{r}\right)}^2}=\frac{\mathrm{GMdm}}{9{\mathrm{r}}^2}$

This force can be resolved into x and y components as shown. Now consider another element at B. We find that  components (namely $\mathrm{dFcos}\mathrm{\theta }$) add up but y-components (namely of $\mathrm{dFsin}\mathrm{\theta }$) cancel each other. Hence the x components of forces on all elements add up and y-components cancel. Hence, the total force exerted by the sphere on the ring is

$\begin{array}{l}\mathrm{F}=\oint \mathrm{dFcos}\mathrm{\theta }=\oint \frac{\mathrm{GMdm}}{9{\mathrm{r}}^2}\times \frac{2\sqrt{2}}{3}\\ =\frac{2\sqrt{2}\mathrm{GM}}{27{\mathrm{r}}^2}\oint \mathrm{dm}=\frac{2\sqrt{2}\mathrm{GMm}}{27{\mathrm{r}}^2}\end{array}$

So the correct choice is (1).