There are two circle . The length of arc make angle at the centre `60^(@) and 75^(@)` . If the length of arc is same then find the ratio of their radius.

Answer

To solve the problem, we will 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 calculated using the formula:
\[ L = \theta \times R \]
where \( \theta \) is the angle in radians and \( R \) is the radius of the circle.

### Step 2: Convert angles from degrees to radians
Since the angles given in the problem are in degrees, we need to convert them to radians. The conversion formula is:
\[ \text{radians} = \frac{\text{degrees} \times \pi}{180} \]
For \( \theta_1 = 60^\circ \):
\[ \theta_1 = \frac{60 \times \pi}{180} = \frac{\pi}{3} \]
For \( \theta_2 = 75^\circ \):
\[ \theta_2 = \frac{75 \times \pi}{180} = \frac{5\pi}{12} \]

### Step 3: Set up the equation for the lengths of the arcs
Since the lengths of the arcs are the same, we can write:
\[ L_1 = L_2 \]
Using the formula for arc length, we can express this as:
\[ \theta_1 \times R_1 = \theta_2 \times R_2 \]

### Step 4: Rearrange the equation to find the ratio of the radii
From the equation \( \theta_1 \times R_1 = \theta_2 \times R_2 \), we can rearrange it to find the ratio of the radii:
\[ \frac{R_1}{R_2} = \frac{\theta_2}{\theta_1} \]

### Step 5: Substitute the values of \( \theta_1 \) and \( \theta_2 \)
Now substituting the values we found:
\[ \frac{R_1}{R_2} = \frac{\frac{5\pi}{12}}{\frac{\pi}{3}} \]

### Step 6: Simplify the ratio
To simplify:
\[ \frac{R_1}{R_2} = \frac{5\pi}{12} \times \frac{3}{\pi} = \frac{5 \times 3}{12} = \frac{15}{12} = \frac{5}{4} \]

### Conclusion
Thus, the ratio of the radii \( R_1 : R_2 \) is \( 5 : 4 \).

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