Let `x^2/a^2+y^2/b^2=1 and x^2/A^2-y^2/B^2=1` be confocal `(a > A and a> b)` having the foci at `s_1 and S_2,` respectively. If P is their point of intersection, then `S_1 P and S_2 P` are the roots of quadratic equation

Options

A. (a) `x^(2)+2Ax+(a^(2)-A^(2))=0`
B. (b) `x^(2)+2ax+(a^(2)-A^(2))=0`
C. (c) `x^(2)-2Ax+(a^(2)+A^(2))=0`
D. (d) `x^(2)-2ax+(a^(2)-A^(2))=0`

Explanation

To solve the problem, we need to find the roots of the quadratic equation formed by the distances \( S_1P \) and \( S_2P \) from the foci \( S_1 \) and \( S_2 \) to the point of intersection \( P \) of the given ellipse and hyperbola.

### Step-by-Step Solution:

1. **Identify the Equations**:
The equations given are:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad \text{(Ellipse)} \]
\[ \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1 \quad \text{(Hyperbola)} \]

2. **Points of Intersection**:
The point \( P \) is the intersection of the ellipse and hyperbola. This point satisfies both equations.

3. **Distance from Foci**:
The foci of the ellipse are located at \( S_1(a, 0) \) and \( S_1(-a, 0) \).
The foci of the hyperbola are at \( S_2(A, 0) \) and \( S_2(-A, 0) \).

4. **Using Properties of Ellipse**:
For the ellipse, the sum of the distances from the foci to any point on the ellipse is constant:
\[ S_1P + S_2P = 2a \]

5. **Using Properties of Hyperbola**:
For the hyperbola, the difference of the distances from the foci to any point on the hyperbola is constant:
\[ S_1P - S_2P = 2A \]

6. **Setting Up the Equations**:
Let \( S_1P = x \) and \( S_2P = y \). We have the following system of equations:
\[ x + y = 2a \quad \text{(1)} \]
\[ x - y = 2A \quad \text{(2)} \]

7. **Solving the System**:
To find \( x \) and \( y \), we can add and subtract the equations:
- Adding (1) and (2):
\[ (x + y) + (x - y) = 2a + 2A \implies 2x = 2a + 2A \implies x = a + A \]
- Subtracting (2) from (1):
\[ (x + y) - (x - y) = 2a - 2A \implies 2y = 2a - 2A \implies y = a - A \]

8. **Roots of the Quadratic Equation**:
The distances \( S_1P \) and \( S_2P \) are \( a + A \) and \( a - A \), respectively. The roots of the quadratic equation can be expressed as:
\[ t^2 - (S_1P + S_2P)t + (S_1P \cdot S_2P) = 0 \]
Substituting the values:
\[ t^2 - (2a)t + ((a + A)(a - A)) = 0 \]
Simplifying the product:
\[ (a + A)(a - A) = a^2 - A^2 \]
Thus, the quadratic equation becomes:
\[ t^2 - 2at + (a^2 - A^2) = 0 \]

### Final Quadratic Equation:
The final quadratic equation is:
\[ t^2 - 2at + (a^2 - A^2) = 0 \]