The lengths of the tangents from the points A and B to a circle are `l_(1)` and `l_(2)` respectively. If points are conjugate with respect to the circle, then `AB^(2)=`

Options

A. `l_(1)+l_(2)`
B. `l_(1)^(2)+l_(2)^(2)`
C. `|l_(1)^(2)-l_(2)^(2)|`
D. none of these

Explanation

To solve the problem step by step, we need to derive the expression for \( AB^2 \) given that points A and B are conjugate with respect to a circle.

### Step-by-Step Solution:

1. **Understanding the Circle**:
We consider a circle with the equation \( x^2 + y^2 = a^2 \). Here, \( a \) is the radius of the circle.

2. **Identifying Points A and B**:
Let the coordinates of point A be \( (x_1, y_1) \) and the coordinates of point B be \( (x_2, y_2) \).

3. **Finding Lengths of Tangents**:
The lengths of the tangents from points A and B to the circle are given by:
\[ l_1^2 = x_1^2 + y_1^2 - a^2 \]
\[ l_2^2 = x_2^2 + y_2^2 - a^2 \]

4. **Using the Conjugate Points Condition**:
Since points A and B are conjugate with respect to the circle, the following relationship holds:
\[ x_1 x_2 + y_1 y_2 = a^2 \]

5. **Finding \( AB^2 \)**:
The distance \( AB \) can be expressed as:
\[ AB^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2 \]
Expanding this expression gives:
\[ AB^2 = x_1^2 + y_2^2 + y_1^2 + y_2^2 - 2(x_1 x_2 + y_1 y_2) \]

6. **Substituting the Conjugate Condition**:
We substitute \( x_1 x_2 + y_1 y_2 = a^2 \) into the equation:
\[ AB^2 = x_1^2 + y_1^2 + x_2^2 + y_2^2 - 2a^2 \]

7. **Relating to Tangent Lengths**:
From the tangent lengths, we know:
\[ l_1^2 = x_1^2 + y_1^2 - a^2 \quad \text{and} \quad l_2^2 = x_2^2 + y_2^2 - a^2 \]
Therefore, we can express \( x_1^2 + y_1^2 \) and \( x_2^2 + y_2^2 \) in terms of \( l_1^2 \) and \( l_2^2 \):
\[ x_1^2 + y_1^2 = l_1^2 + a^2 \quad \text{and} \quad x_2^2 + y_2^2 = l_2^2 + a^2 \]

8. **Substituting Back**:
Substitute these into the expression for \( AB^2 \):
\[ AB^2 = (l_1^2 + a^2) + (l_2^2 + a^2) - 2a^2 \]
Simplifying this gives:
\[ AB^2 = l_1^2 + l_2^2 \]

### Final Result:
Thus, the final expression for \( AB^2 \) is:
\[ AB^2 = l_1^2 + l_2^2 \]