If the circle `x^(2)+y^(2)=a^(2)` intersects the hyperbola `xy=c^(2)` in four points
`P(x_(1),y_(1)),Q(x_(2),y_(2)),R(x_(3),y_(3)),S(x_(4),y_(4))`, then

Options

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

Answer

To solve the problem, we need to analyze the intersection points of the circle \( x^2 + y^2 = a^2 \) and the hyperbola \( xy = c^2 \). We will derive the equations for the intersection points and find the relationships among them.

### Step 1: Substitute for \( y \) from the hyperbola equation
From the hyperbola \( xy = c^2 \), we can express \( y \) in terms of \( x \):
\[ y = \frac{c^2}{x} \]

### Step 2: Substitute \( y \) into the circle equation
Now, substitute \( y \) into the circle equation \( x^2 + y^2 = a^2 \):
\[ x^2 + \left(\frac{c^2}{x}\right)^2 = a^2 \]
This simplifies to:
\[ x^2 + \frac{c^4}{x^2} = a^2 \]

### Step 3: Multiply through by \( x^2 \)
To eliminate the fraction, multiply the entire equation by \( x^2 \):
\[ x^4 + c^4 = a^2 x^2 \]

### Step 4: Rearrange into standard polynomial form
Rearranging gives us a polynomial equation:
\[ x^4 - a^2 x^2 + c^4 = 0 \]

### Step 5: Let \( z = x^2 \)
Let \( z = x^2 \) to convert this into a quadratic equation:
\[ z^2 - a^2 z + c^4 = 0 \]

### Step 6: Find the sum and product of the roots
Using Vieta's formulas for the quadratic \( z^2 - a^2 z + c^4 = 0 \):
- The sum of the roots \( z_1 + z_2 = a^2 \)
- The product of the roots \( z_1 z_2 = c^4 \)

### Step 7: Relate back to \( x \)
Since \( z = x^2 \), the roots \( x_1^2, x_2^2, x_3^2, x_4^2 \) correspond to the four intersection points. Thus:
- The sum of the \( x \) coordinates \( x_1 + x_2 + x_3 + x_4 = 0 \) (since the coefficients of \( x^3 \) in the original polynomial is 0).
- The product of the \( x \) coordinates \( x_1 x_2 x_3 x_4 = c^4 \).

### Step 8: Repeat for \( y \)
Now, we can perform a similar analysis for \( y \):
1. Substitute \( x = \frac{c^2}{y} \) into the circle equation.
2. This leads to a similar polynomial in terms of \( y \):
\[ y^4 - a^2 y^2 + c^4 = 0 \]
3. Again, using Vieta's formulas:
- The sum of the \( y \) coordinates \( y_1 + y_2 + y_3 + y_4 = 0 \).
- The product of the \( y \) coordinates \( y_1 y_2 y_3 y_4 = c^4 \).

### Conclusion
Thus, we conclude that:
- The sum of the \( x \) coordinates is \( 0 \).
- The product of the \( x \) coordinates is \( c^4 \).
- The sum of the \( y \) coordinates is \( 0 \).
- The product of the \( y \) coordinates is \( c^4 \).