The locus of the mid point of the chords of the circle  $C_1:{(x-4)}^2+{(y-5)}^2=4$ which subtend an angle  ${\theta }_i$ at the centre of the circle $C_1$, is a circle of radius $r_i$ . If ${\theta }_1=\frac{\pi }{3},{\theta }_3=\frac{2\pi }{3}$ and  $r_1^2=r_2^2+r_3^2$, then  ${\theta }_2$ is equal to