If the arcs of the same length in two circles subtend angles of `60^(@)` and `90^(@)` at the centre, then the ratio of the their radii is :

Options

A. `(1)/(3)`
B. `(1)/(2)`
C. `(3)/(2)`
D. 2

Answer

To solve the problem, we need to find the ratio of the radii of two circles given that the arcs of the same length subtend angles of \(60^\circ\) and \(90^\circ\) at the center of the circles.

### Step-by-Step Solution:

1. **Understanding Arc Length Formula**:
The length of an arc \(L\) in a circle can be calculated using the formula:
\[ L = r \cdot \theta \]
where \(r\) is the radius of the circle and \(\theta\) is the angle in radians.

2. **Convert Degrees to Radians**:
We need to convert the angles from degrees to radians:
- For \(90^\circ\):
\[ \theta_1 = 90^\circ = \frac{\pi}{180} \times 90 = \frac{\pi}{2} \text{ radians} \]
- For \(60^\circ\):
\[ \theta_2 = 60^\circ = \frac{\pi}{180} \times 60 = \frac{\pi}{3} \text{ radians} \]

3. **Setting Up the Equations**:
Let \(R_1\) be the radius of the first circle (subtending \(90^\circ\)) and \(R_2\) be the radius of the second circle (subtending \(60^\circ\)). Since the lengths of the arcs are equal, we can write:
\[ L_1 = R_1 \cdot \theta_1 \quad \text{and} \quad L_2 = R_2 \cdot \theta_2 \]
Given that \(L_1 = L_2\), we have:
\[ R_1 \cdot \frac{\pi}{2} = R_2 \cdot \frac{\pi}{3} \]

4. **Canceling \(\pi\)**:
We can cancel \(\pi\) from both sides:
\[ R_1 \cdot \frac{1}{2} = R_2 \cdot \frac{1}{3} \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ \frac{R_2}{R_1} = \frac{1/2}{1/3} = \frac{1}{2} \cdot \frac{3}{1} = \frac{3}{2} \]

6. **Conclusion**:
Therefore, the ratio of the radii \(R_2\) to \(R_1\) is:
\[ \frac{R_2}{R_1} = \frac{3}{2} \]

### Final Answer:
The ratio of the radii is \(\frac{3}{2}\).

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