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

Let \( l \) be the length of the arc in both circles.
Let \( r_1 \) and \( r_2 \) be the radii of the two circles.
Let \( \theta_1 \) and \( \theta_2 \) be the angles subtended by the arc at the center of the respective circles.
We are given that \( \theta_1 = 30^\circ \) and \( \theta_2 = 78^\circ \).
The length of an arc is given by the formula \( l = r\theta \), where \( \theta \) is in radians.

We have:

\[ l = r_1 \theta_1 = r_2 \theta_2 \]

We need to convert the angles to radians:

\[ \theta_1 = 30^\circ = \frac{30\pi}{180} = \frac{\pi}{6} \text{ radians} \] \[ \theta_2 = 78^\circ = \frac{78\pi}{180} = \frac{13\pi}{30} \text{ radians} \]

Now we have:

\[ l = r_1 \left(\frac{\pi}{6}\right) = r_2 \left(\frac{13\pi}{30}\right) \]

We want to find the ratio \( \frac{r_1}{r_2} \):

\[ \frac{r_1}{r_2} = \frac{\frac{13\pi}{30}}{\frac{\pi}{6}} = \frac{13\pi}{30} \times \frac{6}{\pi} = \frac{13 \times 6}{30} = \frac{13}{5} \]

Therefore, the ratio of their radii is \( \frac{13}{5} \).