If in two circles, arcs of the same length subtend angles $30^{\circ}$ and $78^{\circ}$ at the centre, then the ratio of their radii is

Let the radii of the two circles be $r_1$ and $r_2$. Let an arc of length $l$ subtend an angle of $30^{\circ}$ at the centre of the circle of radius $r_1$, while let an arc of length $l$ subtend an angle of $78^{\circ}$ at the centre of the circle of radius $r_2$. Now, $30^{\circ}=\frac{\pi}{6}$ radian and $78^{\circ}=\frac{13 \pi}{30}$ radian We know that in a circle of radius $r$ unit, if an arc of length $l$ unit subtend an angle $\theta$ radian at the centre, then $\theta=\frac{l}{r}$ or $l=r \theta$ $\therefore \quad l=\frac{r_1 \pi}{6}$ and $l=\frac{r_2 13 \pi}{30}$ $\Rightarrow \quad \frac{r_1 \pi \pi}{6}=\frac{r_2 13 \pi}{30}$ $\Rightarrow \quad r_1=r_2 \frac{13}{5}$ $\Rightarrow \quad \frac{r_1}{r_2}=\frac{13}{5}$