If radii are 2, `sqrt2` and distance between centres is `sqrt2` then the angle between the circles is

Options

A. `pi/6`
B. `pi/3`
C. `pi/2`
D. `(3pi)/4`

Answer

To solve the problem of finding the angle between two circles given their radii and the distance between their centers, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Let the radius of the first circle \( r_1 = 2 \).
- Let the radius of the second circle \( r_2 = \sqrt{2} \).
- Let the distance between the centers of the circles \( d = \sqrt{2} \).

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

3. **Substitute the Values:**
- Calculate \( d^2 \):
\[ d^2 = (\sqrt{2})^2 = 2 \]
- Calculate \( r_1^2 \):
\[ r_1^2 = 2^2 = 4 \]
- Calculate \( r_2^2 \):
\[ r_2^2 = (\sqrt{2})^2 = 2 \]

4. **Plug Values into the Formula:**
\[ \cos \theta = \frac{2 - 4 - 2}{2 \cdot 2 \cdot \sqrt{2}} \]
Simplifying the numerator:
\[ 2 - 4 - 2 = -4 \]
Thus, we have:
\[ \cos \theta = \frac{-4}{4\sqrt{2}} = \frac{-1}{\sqrt{2}} = -\frac{\sqrt{2}}{2} \]

5. **Find the Angle \( \theta \):**
The value \( \cos \theta = -\frac{\sqrt{2}}{2} \) corresponds to angles in the second quadrant. The angle that gives this cosine value is:
\[ \theta = \frac{3\pi}{4} \]

6. **Conclusion:**
Therefore, the angle between the two circles is:
\[ \theta = \frac{3\pi}{4} \]

### Final Answer:
The angle between the circles is \( \frac{3\pi}{4} \).

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