If A and B are conjugate points w.r.t to circle `x^(2)+y^(2)=r^(2)` then `OA^(2)+OB^(2)=`

Options

A. `AB^(2)-r^(2)`
B. `AB^(2)+r^(2)`
C. `AB^(2)+2r^(2)`
D. `AB^(2)-2r^(2)`

Answer

To solve the problem, we need to find the expression for \( OA^2 + OB^2 \) given that points A and B are conjugate points with respect to the circle defined by the equation \( x^2 + y^2 = r^2 \).

### Step-by-Step Solution:

1. **Understand the Circle and Points**:
The equation of the circle is \( x^2 + y^2 = r^2 \). Let the coordinates of points A and B be \( A(x_1, y_1) \) and \( B(x_2, y_2) \).

2. **Conjugate Points Condition**:
For points A and B to be conjugate points with respect to the circle, they must satisfy the condition \( S_{12} = 0 \), where \( S \) is the equation of the circle. This condition can be expressed as:
\[ S_{12} = x_1 x_2 + y_1 y_2 = r^2 \]
(This is our equation 1.)

3. **Calculate \( OA^2 \) and \( OB^2 \)**:
- The distance \( OA \) from the origin \( O(0, 0) \) to point A is given by:
\[ OA^2 = (x_1 - 0)^2 + (y_1 - 0)^2 = x_1^2 + y_1^2 \]
- Similarly, the distance \( OB \) from the origin to point B is:
\[ OB^2 = (x_2 - 0)^2 + (y_2 - 0)^2 = x_2^2 + y_2^2 \]

4. **Combine the Results**:
Now, we can express \( OA^2 + OB^2 \):
\[ OA^2 + OB^2 = x_1^2 + y_1^2 + x_2^2 + y_2^2 \]

5. **Express \( AB^2 \)**:
The distance \( AB \) between points A and B is given by:
\[ AB^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 \]
Expanding this gives:
\[ AB^2 = (x_2^2 - 2x_1x_2 + x_1^2) + (y_2^2 - 2y_1y_2 + y_1^2) \]
This simplifies to:
\[ AB^2 = x_1^2 + y_1^2 + x_2^2 + y_2^2 - 2(x_1x_2 + y_1y_2) \]

6. **Substitute the Conjugate Condition**:
From equation 1, we know \( x_1x_2 + y_1y_2 = r^2 \). Substituting this into the equation for \( AB^2 \):
\[ AB^2 = x_1^2 + y_1^2 + x_2^2 + y_2^2 - 2r^2 \]

7. **Rearranging the Equation**:
Rearranging gives:
\[ x_1^2 + y_1^2 + x_2^2 + y_2^2 = AB^2 + 2r^2 \]

8. **Final Result**:
Therefore, we can conclude that:
\[ OA^2 + OB^2 = AB^2 + 2r^2 \]

### Final Answer:
Thus, the expression for \( OA^2 + OB^2 \) is:
\[ OA^2 + OB^2 = AB^2 + 2r^2 \]