From figure $\left(\frac{\mathrm{R}}{\sqrt{2}}, \frac{\mathrm{R}}{\sqrt{2}}\right)$ is the coordinate of the rod only in absence of the ring,

When the ring is also introduced then ${\mathrm{r}}_{\mathrm{c}\mathrm{m}}=\frac{{\mathrm{m}}_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{r}\mathrm{d}} {\mathrm{r}}_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{r}\mathrm{d}}+{\mathrm{m}}_{\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}} {\mathrm{r}}_{\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}}{{\mathrm{m}}_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{r}\mathrm{d}}+{\mathrm{m}}_{\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}}$

Hence coordinate of the center of mass of system ${\mathrm{r}}_{\mathrm{c}\mathrm{m}}=\frac{{\mathrm{m}}_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{r}\mathrm{d}} {\mathrm{r}}_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{r}\mathrm{d}}}{{\mathrm{m}}_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{r}\mathrm{d}}+{\mathrm{m}}_{\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}}$

Hence the value of coordinate will decrease and will not be $\left(\frac{\mathrm{R}}{\sqrt{2}}, \frac{\mathrm{R}}{\sqrt{2}}\right)$, hence the answer is B