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

Given:

Two circles have arcs of equal length that subtend angles of 30° and 78° at their centers.

Step 1: Recall the arc length formula.

Arc length \( l = r \theta \), where \( r \) is radius and \( \theta \) is in radians.

Step 2: Convert angles to radians.

First angle: \( 30° = \frac{\pi}{6} \) radians

Second angle: \( 78° = \frac{13\pi}{30} \) radians

Step 3: Set arc lengths equal.

\( r_1 \cdot \frac{\pi}{6} = r_2 \cdot \frac{13\pi}{30} \)

Step 4: Solve for the ratio \( \frac{r_1}{r_2} \).

\( \frac{r_1}{r_2} = \frac{13\pi/30}{\pi/6} = \frac{13}{5} \)

Option Analysis:

(A) \( \frac{5}{13} \) - Inverse of correct ratio

(B) \( \frac{13}{5} \) - Correct

(C) \( \frac{13}{4} \) - Incorrect

(D) \( \frac{4}{1} \) - Incorrect

Final Answer: \(\boxed{B}\)

Answer 2

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: $\theta_1 = 30^\circ = \frac{\pi}{6}, \quad \theta_2 = 78^\circ = \frac{13\pi}{30}$. $\frac{r_1}{r_2} = \frac{\frac{13\pi}{30}}{\frac{\pi}{6}} = \frac{13}{5}$.