If the circle `x^2+y^2=a^2` intersects the hyperbola `x y=c^2` at four points `P(x_1, y_1),Q(x_2, y_2),R(x_3, y_3),` and `S(x_4, y_4),` then `x_1+x_2+x_3+x_4=0` `y_1+y_2+y_3+y_4=0` `x_1x_2x_3x_4=C^4` `y_1y_2y_3y_4=C^4`

Options

A. `x_(1)+x_(2)+x_(3)+x_(4)=0`
B. `y_(1)+y_(2)+y_(3)+y_(4)=0`
C. `x_(1)x_(2)x_(3)x_(4)=c^(4)`
D. `y_(1)y_(2)y_(3)y_(4)=c^(4)`

Explanation

To solve the problem, we need to analyze the intersection of the circle \(x^2 + y^2 = a^2\) and the hyperbola \(xy = c^2\). The points of intersection are denoted as \(P(x_1, y_1)\), \(Q(x_2, y_2)\), \(R(x_3, y_3)\), and \(S(x_4, y_4)\).

### Step-by-step Solution:

1. **Substitute \(y\) from the hyperbola into the circle's equation**:
We know from the hyperbola that \(y = \frac{c^2}{x}\). Substitute this into the circle's equation:
\[ x^2 + \left(\frac{c^2}{x}\right)^2 = a^2 \]
This simplifies to:
\[ x^2 + \frac{c^4}{x^2} = a^2 \]

2. **Multiply through by \(x^2\) to eliminate the fraction**:
\[ x^4 - a^2x^2 + c^4 = 0 \]
This is a polynomial equation in terms of \(x^2\).

3. **Let \(u = x^2\)**:
The equation can be rewritten as:
\[ u^2 - a^2u + c^4 = 0 \]
This is a quadratic equation in \(u\).

4. **Find the sum and product of the roots**:
By Vieta's formulas, for the quadratic equation \(u^2 - a^2u + c^4 = 0\):
- The sum of the roots \(u_1 + u_2 = a^2\)
- The product of the roots \(u_1 u_2 = c^4\)

5. **Relate back to \(x\)**:
Since \(u = x^2\), we can say:
\[ x_1^2 + x_2^2 + x_3^2 + x_4^2 = 2(a^2) \]
However, we need to find \(x_1 + x_2 + x_3 + x_4\). The roots of the polynomial \(x^4 - a^2x^2 + c^4 = 0\) have a symmetric property, leading to:
\[ x_1 + x_2 + x_3 + x_4 = 0 \]

6. **For \(y\)**:
Using the same substitution \(y = \frac{c^2}{x}\), we can derive a similar polynomial for \(y\):
\[ y^4 - a^2y^2 + c^4 = 0 \]
By Vieta's formulas again, we find:
- The sum of the roots \(y_1 + y_2 + y_3 + y_4 = 0\)
- The product \(y_1 y_2 y_3 y_4 = c^4\)

### Conclusion:
From the analysis, we conclude:
- \(x_1 + x_2 + x_3 + x_4 = 0\)
- \(y_1 + y_2 + y_3 + y_4 = 0\)
- \(x_1 x_2 x_3 x_4 = c^4\)
- \(y_1 y_2 y_3 y_4 = c^4\)

Thus, all the given statements in the problem are correct.