If the length of the tangent from two points A,B to a circle are 6,7 respectively. If A,B are conjugate points then AB=

Options

A. `85`
B. `42`
C. `sqrt(85)`
D. `sqrt(13)`

Answer

To solve the problem, we need to find the distance \( AB \) between the two conjugate points \( A \) and \( B \) given the lengths of the tangents from these points to a circle.

### Step-by-Step Solution:

1. **Identify the lengths of the tangents**:
- The length of the tangent from point \( A \) to the circle is given as \( L_1 = 6 \).
- The length of the tangent from point \( B \) to the circle is given as \( L_2 = 7 \).

2. **Use the formula for the distance between conjugate points**:
- The distance \( AB \) between two conjugate points \( A \) and \( B \) can be calculated using the formula:
\[ AB = \sqrt{L_1^2 + L_2^2} \]

3. **Substitute the values of \( L_1 \) and \( L_2 \)**:
- Substitute \( L_1 = 6 \) and \( L_2 = 7 \) into the formula:
\[ AB = \sqrt{6^2 + 7^2} \]

4. **Calculate \( L_1^2 \) and \( L_2^2 \)**:
- Calculate \( 6^2 = 36 \) and \( 7^2 = 49 \).

5. **Add the squares**:
- Now, add the squares:
\[ 36 + 49 = 85 \]

6. **Take the square root**:
- Finally, take the square root of the sum:
\[ AB = \sqrt{85} \]

### Final Answer:
Thus, the distance \( AB \) is \( \sqrt{85} \).

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