The locus of the mid-points 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 $\theta_{1}=\dfrac{\pi}{3}$, $\theta_{3}=\dfrac{2\pi}{3}$ and $r_{1}^{2}=r_{2}^{2}+r_{3}^{2}$, then $\theta_{2}$ is equal to: