If the arcs of the same length in two circles subtend angles `65^(@)` and `110^(@)` at the centre then the ratio of the radii of the circles is

Options

A. `22:13`
B. `11:13`
C. `22:15`
D. `21:13`

Answer

To solve the problem of finding the ratio of the radii of two circles given that the arcs of the same length subtend angles of \(65^\circ\) and \(110^\circ\) at the center, we can follow these steps:

### Step 1: Understand the relationship between arc length, radius, and angle
The length of an arc \(L\) in a circle can be expressed using the formula:
\[ L = r \cdot \theta \]
where \(r\) is the radius of the circle and \(\theta\) is the angle in radians.

### Step 2: Convert angles from degrees to radians
To use the formula, we need to convert the angles from degrees to radians. The conversion is done using the relation:
\[ \text{radians} = \frac{\pi}{180} \times \text{degrees} \]
For \(65^\circ\):
\[ \theta_1 = \frac{\pi}{180} \times 65 \]
For \(110^\circ\):
\[ \theta_2 = \frac{\pi}{180} \times 110 \]

### Step 3: Set up the equations for the arc lengths
Since the lengths of the arcs \(L_1\) and \(L_2\) are equal, we can write:
\[ L_1 = r_1 \cdot \theta_1 \quad \text{and} \quad L_2 = r_2 \cdot \theta_2 \]
Given \(L_1 = L_2\), we have:
\[ r_1 \cdot \theta_1 = r_2 \cdot \theta_2 \]

### Step 4: Rearrange the equation to find the ratio of the radii
Rearranging the equation gives us:
\[ \frac{r_1}{r_2} = \frac{\theta_2}{\theta_1} \]

### Step 5: Substitute the values of \(\theta_1\) and \(\theta_2\)
Substituting the values we calculated:
\[ \frac{r_1}{r_2} = \frac{\frac{\pi}{180} \times 110}{\frac{\pi}{180} \times 65} \]
The \(\frac{\pi}{180}\) cancels out:
\[ \frac{r_1}{r_2} = \frac{110}{65} \]

### Step 6: Simplify the ratio
Now, we simplify \(\frac{110}{65}\):
\[ \frac{110}{65} = \frac{22}{13} \]

### Final Answer
Thus, the ratio of the radii of the two circles is:
\[ \frac{r_1}{r_2} = \frac{22}{13} \]