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 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:
