Let `x^2 + y^2 = r^2` and `xy = 1` intersect at `A & B` in first quadrant, If `AB = sqrt14` then find the value of `r.`

Answer

To solve the problem, we need to find the value of \( r \) given the equations \( x^2 + y^2 = r^2 \) and \( xy = 1 \), and the distance \( AB = \sqrt{14} \) between the points of intersection \( A \) and \( B \) in the first quadrant.

### Step-by-Step Solution:

1. **Substitute \( y \) in terms of \( x \)**:
Since \( xy = 1 \), we can express \( y \) as:
\[ y = \frac{1}{x} \]

2. **Substitute \( y \) into the first equation**:
Substitute \( y \) into the equation \( x^2 + y^2 = r^2 \):
\[ x^2 + \left(\frac{1}{x}\right)^2 = r^2 \]
This simplifies to:
\[ x^2 + \frac{1}{x^2} = r^2 \]

3. **Multiply through by \( x^2 \)**:
To eliminate the fraction, multiply the entire equation by \( x^2 \):
\[ x^4 + 1 = r^2 x^2 \]
Rearranging gives:
\[ x^4 - r^2 x^2 + 1 = 0 \]

4. **Let \( z = x^2 \)**:
Let \( z = x^2 \). The equation becomes:
\[ z^2 - r^2 z + 1 = 0 \]

5. **Use the quadratic formula**:
The roots of this quadratic equation are given by:
\[ z = \frac{r^2 \pm \sqrt{(r^2)^2 - 4}}{2} \]

6. **Sum and product of roots**:
The sum of the roots \( z_1 + z_2 = r^2 \) and the product \( z_1 z_2 = 1 \). Therefore, we have:
\[ z_1 z_2 = 1 \implies z_1 z_2 = x_1^2 x_2^2 = (x_1 x_2)^2 = 1 \implies x_1 x_2 = 1 \]

7. **Difference of roots**:
The difference of the roots can be calculated as:
\[ z_1 - z_2 = \sqrt{(z_1 + z_2)^2 - 4z_1 z_2} = \sqrt{r^4 - 4} \]

8. **Distance \( AB \)**:
The distance \( AB \) can be expressed as:
\[ AB = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \]
Since \( y_1 = \frac{1}{x_1} \) and \( y_2 = \frac{1}{x_2} \):
\[ AB = \sqrt{(x_1 - x_2)^2 + \left(\frac{1}{x_1} - \frac{1}{x_2}\right)^2} \]
This simplifies to:
\[ AB = \sqrt{(x_1 - x_2)^2 + \left(\frac{x_2 - x_1}{x_1 x_2}\right)^2} = \sqrt{(x_1 - x_2)^2 \left(1 + \frac{1}{(x_1 x_2)^2}\right)} \]
Since \( x_1 x_2 = 1 \):
\[ AB = |x_1 - x_2| \sqrt{2} \]

9. **Setting the distance equal to \( \sqrt{14} \)**:
We know \( AB = \sqrt{14} \):
\[ |x_1 - x_2| \sqrt{2} = \sqrt{14} \]
Thus,
\[ |x_1 - x_2| = \sqrt{7} \]

10. **Relate \( |x_1 - x_2| \) to \( r \)**:
From the earlier step:
\[ |x_1 - x_2| = \sqrt{r^4 - 4} \]
Therefore,
\[ \sqrt{r^4 - 4} = \sqrt{7} \]

11. **Square both sides**:
\[ r^4 - 4 = 7 \implies r^4 = 11 \]

12. **Find \( r \)**:
Taking the fourth root:
\[ r = \sqrt[4]{11} \]

### Final Answer:
\[ r = \sqrt[4]{11} \]