The locus of the mid points of the chords of the circle ${\mathrm{C}}_1:(\mathrm{x}-4{)}^2+(\mathrm{y}-5{)}^2=4$ which subtend
an angle ${\theta }_1$ at the centre of the circle ${\mathrm{C}}_1$ , is a circle of radius $r_1$. If ${\mathrm{\theta }}_1=\frac{\mathrm{\pi }}{3},{\mathrm{\theta }}_3=\frac{2\mathrm{\pi }}{3}$ and ${\mathrm{r}}_1^2={\mathrm{r}}_2^2+{\mathrm{r}}_3^2$ , then ${\mathrm{\theta }}_2$ is equal to