If the circle `x^(2) + y^(2) = a^(2)` intersects the hyperbola xy = 25 in four points, then find the product of the ordinates of these points.

Answer

To find the product of the ordinates of the points where the circle \( x^2 + y^2 = a^2 \) intersects the hyperbola \( xy = 25 \), we can follow these steps:

### Step 1: Substitute the hyperbola equation into the circle equation
We know from the hyperbola that \( y = \frac{25}{x} \). We can substitute this into the circle equation:

\[ x^2 + y^2 = a^2 \implies x^2 + \left(\frac{25}{x}\right)^2 = a^2 \]

### Step 2: Simplify the equation
Substituting \( y \) gives us:

\[ x^2 + \frac{625}{x^2} = a^2 \]

### Step 3: Multiply through by \( x^2 \) to eliminate the fraction
To eliminate the fraction, multiply the entire equation by \( x^2 \):

\[ x^4 - a^2 x^2 + 625 = 0 \]

### Step 4: Let \( z = x^2 \)
Let \( z = x^2 \). The equation becomes:

\[ z^2 - a^2 z + 625 = 0 \]

### Step 5: Use the quadratic formula to find the roots
Using the quadratic formula \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

\[ z = \frac{a^2 \pm \sqrt{(a^2)^2 - 4 \cdot 1 \cdot 625}}{2 \cdot 1} \]

### Step 6: Calculate the product of the roots
The product of the roots \( z_1 \) and \( z_2 \) of the quadratic equation \( z^2 - a^2 z + 625 = 0 \) is given by:

\[ z_1 z_2 = \frac{625}{1} = 625 \]

Since \( z = x^2 \), we have:

\[ x_1^2 x_2^2 = 625 \]

### Step 7: Find the product of the ordinates
The ordinates corresponding to \( x_1 \) and \( x_2 \) are:

\[ y_1 = \frac{25}{x_1}, \quad y_2 = \frac{25}{x_2} \]

Thus, the product of the ordinates \( y_1 y_2 \) is:

\[ y_1 y_2 = \left(\frac{25}{x_1}\right) \left(\frac{25}{x_2}\right) = \frac{625}{x_1 x_2} \]

Using the relationship \( x_1 x_2 = \sqrt{z_1 z_2} = \sqrt{625} = 25 \):

\[ y_1 y_2 = \frac{625}{25} = 25 \]

### Conclusion
The product of the ordinates of the four intersection points is:

\[ \text{Product of ordinates} = 25 \]