If radii of two circles are 4 and 3 and distance between centres is `sqrt37`, then angle between the circles is

Options

A. `30^@`
B. `45^@`
C. `60^@`
D. `90^@`

Answer

To find the angle between two circles given their radii and the distance between their centers, we can use the formula derived from the cosine rule. Here’s how we can solve the problem step by step:

### Step-by-Step Solution

1. **Identify the Given Values**:
- Radius of the first circle, \( r_1 = 4 \)
- Radius of the second circle, \( r_2 = 3 \)
- Distance between the centers of the circles, \( d = \sqrt{37} \)

2. **Use the Cosine Rule**:
The angle \( \theta \) between the two circles can be found using the formula:
\[ \cos \theta = \frac{d^2 - r_1^2 - r_2^2}{2 r_1 r_2} \]

3. **Calculate \( d^2 \)**:
\[ d^2 = (\sqrt{37})^2 = 37 \]

4. **Calculate \( r_1^2 \) and \( r_2^2 \)**:
\[ r_1^2 = 4^2 = 16 \]
\[ r_2^2 = 3^2 = 9 \]

5. **Substitute the Values into the Formula**:
\[ \cos \theta = \frac{37 - 16 - 9}{2 \cdot 4 \cdot 3} \]

6. **Simplify the Numerator**:
\[ 37 - 16 - 9 = 12 \]

7. **Calculate the Denominator**:
\[ 2 \cdot 4 \cdot 3 = 24 \]

8. **Final Calculation for \( \cos \theta \)**:
\[ \cos \theta = \frac{12}{24} = \frac{1}{2} \]

9. **Find the Angle \( \theta \)**:
Since \( \cos \theta = \frac{1}{2} \), we know that:
\[ \theta = 60^\circ \]

### Conclusion
The angle between the two circles is \( 60^\circ \).