1) From the centre of mass, we get

${\mathrm{M}}_1{\mathrm{R}}_1={\mathrm{M}}_2{\mathrm{R}}_2------1$

2) Force acting on both starts i.e. gravitational force and centripetal force are balanced.

$\frac{{\mathrm{GM}}_1{\mathrm{M}}_2}{{\mathrm{R}}^2}=\frac{{\mathrm{M}}_1{\mathrm{V}}_1^2}{{\mathrm{R}}_1}=\frac{{\mathrm{M}}_1{\mathrm{V}}_1^2}{{\mathrm{R}}_1}------2$

As we know, ${\mathrm{V}}_1=\frac{2{\mathrm{\pi R}}_1}{{\mathrm{T}}_1}\&{\mathrm{V}}_2=\frac{2{\mathrm{\pi R}}_2}{{\mathrm{T}}_2}$

Substituting equation 3 in equation 2, we get 

$\frac{{\mathrm{M}}_1{\mathrm{R}}_1}{{\left({\mathrm{T}}_1\right)}^2}=\frac{{\mathrm{M}}_1{\mathrm{R}}_2}{{\left({\mathrm{T}}_2\right)}^2}------4$

From equation 1, we conclude in eqution 4

${\left({\mathrm{T}}_1\right)}^2={\left({\mathrm{T}}_2\right)}^2 \mathrm{or} {\mathrm{T}}_1={\mathrm{T}}_2$