If the sum of the squares of the distances of the point (x, y) from the points (a, 0) and (-a,0) is `2b^(2)`, then which one of the following is correct ?

Options

A. `x^(2) + a^(2) = b^(2) + y^(2)`
B. `x^(2) + a^(2) = 2b^(2) - y^(2)`
C. `x^(2) - a^(2) = b^(2) + y^(2)`
D. `x^(2) + a^(2) = b^(2) - y^(2)`

Answer

To solve the problem, we need to find the relationship between the coordinates of a point (x, y) and the points (a, 0) and (-a, 0) given that the sum of the squares of the distances from these points equals \(2b^2\).

### Step-by-Step Solution:

1. **Identify the Points**:
- Let point A be (a, 0) and point B be (-a, 0).
- Let point C be (x, y).

2. **Calculate the Distance from Point C to Point A**:
- The distance \(d_{CA}\) from point C to point A is given by the distance formula:
\[ d_{CA} = \sqrt{(x - a)^2 + (y - 0)^2} = \sqrt{(x - a)^2 + y^2} \]
- Therefore, the square of the distance \(d_{CA}^2\) is:
\[ d_{CA}^2 = (x - a)^2 + y^2 \]

3. **Calculate the Distance from Point C to Point B**:
- The distance \(d_{CB}\) from point C to point B is:
\[ d_{CB} = \sqrt{(x + a)^2 + (y - 0)^2} = \sqrt{(x + a)^2 + y^2} \]
- Therefore, the square of the distance \(d_{CB}^2\) is:
\[ d_{CB}^2 = (x + a)^2 + y^2 \]

4. **Sum of the Squares of the Distances**:
- According to the problem, the sum of the squares of the distances from point C to points A and B is given by:
\[ d_{CA}^2 + d_{CB}^2 = 2b^2 \]
- Substituting the expressions we found:
\[ (x - a)^2 + y^2 + (x + a)^2 + y^2 = 2b^2 \]

5. **Simplify the Equation**:
- Combine the terms:
\[ (x - a)^2 + (x + a)^2 + 2y^2 = 2b^2 \]
- Expanding the squares:
\[ (x^2 - 2ax + a^2) + (x^2 + 2ax + a^2) + 2y^2 = 2b^2 \]
- Combine like terms:
\[ 2x^2 + 2a^2 + 2y^2 = 2b^2 \]

6. **Divide by 2**:
- Simplifying gives:
\[ x^2 + a^2 + y^2 = b^2 \]

7. **Rearranging the Equation**:
- To express \(x^2 + a^2\) in terms of \(b^2\) and \(y^2\):
\[ x^2 + a^2 = b^2 - y^2 \]

### Conclusion:
The final relationship we derived is:
\[ x^2 + a^2 = b^2 - y^2 \]
This corresponds to option **d**.