If the arcs of the same length in two circle subtend angle(i)`65^(@)` and `110^(@)`
(ii)`60^(@)` and `75^(@)` at the center,
Find the ration of their radii

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 different angles at the center of each circle. We will break this down into two parts as given in the question.

### Part (i): Angles 65° and 110°

1. **Understanding the relationship**: The length of an arc (L) in a circle is given by the formula:
\[ L = r \theta \]
where \( r \) is the radius and \( \theta \) is the angle in radians. However, since we are given angles in degrees, we will convert them to radians when necessary.

2. **Setting up the equation**: For the two circles, we can denote:
- \( L_1 = R_1 \cdot \theta_1 \) for the first circle (where \( \theta_1 = 65^\circ \))
- \( L_2 = R_2 \cdot \theta_2 \) for the second circle (where \( \theta_2 = 110^\circ \))

Since the lengths of the arcs are equal, we have:
\[ R_1 \cdot \theta_1 = R_2 \cdot \theta_2 \]

3. **Rearranging the equation**: We can rearrange this to find the ratio of the radii:
\[ \frac{R_1}{R_2} = \frac{\theta_2}{\theta_1} \]

4. **Substituting the angles**: Now substituting the angles:
\[ \frac{R_1}{R_2} = \frac{110}{65} \]

5. **Simplifying the ratio**: To simplify this ratio:
\[ \frac{R_1}{R_2} = \frac{110 \div 5}{65 \div 5} = \frac{22}{13} \]

### Part (ii): Angles 60° and 75°

1. **Setting up the equation**: Similarly, for the second part, we denote:
- \( L_1 = R_1 \cdot \theta_1 \) for the first circle (where \( \theta_1 = 60^\circ \))
- \( L_2 = R_2 \cdot \theta_2 \) for the second circle (where \( \theta_2 = 75^\circ \))

Again, we have:
\[ R_1 \cdot \theta_1 = R_2 \cdot \theta_2 \]

2. **Rearranging the equation**: Rearranging gives us:
\[ \frac{R_1}{R_2} = \frac{\theta_2}{\theta_1} \]

3. **Substituting the angles**: Now substituting the angles:
\[ \frac{R_1}{R_2} = \frac{75}{60} \]

4. **Simplifying the ratio**: To simplify this ratio:
\[ \frac{R_1}{R_2} = \frac{75 \div 15}{60 \div 15} = \frac{5}{4} \]

### Final Answers:
- For part (i), the ratio of the radii \( R_1 : R_2 = 22 : 13 \).
- For part (ii), the ratio of the radii \( R_1 : R_2 = 5 : 4 \).