Question

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:

• A. $\frac{5}{13}$
• B. $\frac{13}{5}$
• C. $\frac{13}{4}$
• D. $\frac{4}{13}$

Answer 1

The Correct Option is B

The length of an arc is given by $l = r\theta$, where $r$ is the radius and $\theta$ is the angle subtended (in radians).
For the two circles: $\frac{r_1}{r_2} = \frac{\theta_2}{\theta_1}$
Convert degrees to radians: $\theta_1 = 30^\circ = \frac{\pi}{6}, \quad \theta_2 = 78^\circ = \frac{13\pi}{30}$. $\frac{r_1}{r_2} = \frac{\frac{13\pi}{30}}{\frac{\pi}{6}} = \frac{13}{5}$.