Let $r_{1}$ and $r_{2}$ be the radii of the two circles. Given that

${{\theta _1} = {{65}^\circ } = \frac{\pi }{{180}} \times 65 = \frac{{13\pi }}{{36}}\,{\text{ radian }}}$

and    ${{\theta _2} = {{110}^\circ } = \frac{\pi }{{180}} \times 110 = \frac{{22\pi }}{{36}}{\text{ }}\,{\text{radian }}}$

Let $l$ be the length of each of the arc. Then $l=r_{1} \theta_{1}=r_{2} \theta_{2},$ which gives

$\frac{13 \pi}{36} \times r_{1}=\frac{22 \pi}{36} \times r_{2}, \text { i.e., } \frac{r_{1}}{r_{2}}=\frac{22}{13}$

Hence     $r_{1}: r_{2}=22: 13$