A chord AB of a circle `C_1` of radius (square root 3 + 1) cm touches a circel `C_2` of radius (square root 3 -1) cm, then the length of AB is

Options

A. `8 VI cm`
B. `4 root 4 3 cm `
C. `4 VI cm`
D. 2^/1 cm

Answer

To find the length of the chord \( AB \) of circle \( C_1 \) that touches circle \( C_2 \), we can follow these steps:

### Step-by-step Solution:

1. **Identify the Radii of the Circles:**
- The radius of circle \( C_1 \) is \( r_1 = \sqrt{3} + 1 \) cm.
- The radius of circle \( C_2 \) is \( r_2 = \sqrt{3} - 1 \) cm.

2. **Understand the Configuration:**
- Both circles are concentric, meaning they share the same center \( O \).
- Let \( C \) be the midpoint of the chord \( AB \).

3. **Use the Pythagorean Theorem:**
- The distance from the center \( O \) to the chord \( AB \) is \( OC \) which is equal to the radius of circle \( C_2 \), i.e., \( OC = r_2 = \sqrt{3} - 1 \).
- The distance from the center \( O \) to the endpoint of the chord \( A \) is \( OA \), which is equal to the radius of circle \( C_1 \), i.e., \( OA = r_1 = \sqrt{3} + 1 \).

4. **Apply the Pythagorean Theorem:**
- According to the Pythagorean theorem in triangle \( OAC \):
\[ OA^2 = OC^2 + AC^2 \]
- Substitute the values:
\[ (\sqrt{3} + 1)^2 = (\sqrt{3} - 1)^2 + AC^2 \]

5. **Calculate \( OA^2 \) and \( OC^2 \):**
- Calculate \( OA^2 \):
\[ OA^2 = (\sqrt{3} + 1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3} \]
- Calculate \( OC^2 \):
\[ OC^2 = (\sqrt{3} - 1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3} \]

6. **Substitute Back into the Equation:**
- Now substitute \( OA^2 \) and \( OC^2 \) into the equation:
\[ 4 + 2\sqrt{3} = (4 - 2\sqrt{3}) + AC^2 \]
- Rearranging gives:
\[ AC^2 = (4 + 2\sqrt{3}) - (4 - 2\sqrt{3}) = 4\sqrt{3} \]

7. **Find the Length of the Chord \( AB \):**
- Since \( AC \) is half of \( AB \), we have:
\[ AC = \sqrt{4\sqrt{3}} = 2\sqrt[4]{3} \]
- Therefore, the length of the chord \( AB \) is:
\[ AB = 2 \times AC = 2 \times 2\sqrt[4]{3} = 4\sqrt[4]{3} \]

### Final Answer:
The length of the chord \( AB \) is \( 4\sqrt[4]{3} \) cm.