To solve the problem, we need to reflect the graph of the equation \( xy = 1 \) in the line \( y = 2x \) and find the coefficients \( r \), \( s \), and \( t \) in the resulting equation of the form \( 12x^2 + rxy + sy^2 + t = 0 \).
### Step-by-Step Solution:
1. **Identify the Original Equation**:
The original equation is \( xy = 1 \).
2. **Find Points on the Original Curve**:
Let's take a point \( (a, b) \) on the curve \( xy = 1 \). This means \( ab = 1 \).
3. **Reflect the Point Across the Line**:
To reflect the point \( (a, b) \) across the line \( y = 2x \), we need to find the image point \( (α, β) \). The midpoint \( M \) of the segment joining \( (a, b) \) and \( (α, β) \) lies on the line \( y = 2x \).
The midpoint \( M \) is given by:
\[
M = \left( \frac{a + α}{2}, \frac{b + β}{2} \right)
\]
Since \( M \) lies on \( y = 2x \):
\[
\frac{b + β}{2} = 2 \cdot \frac{a + α}{2}
\]
Simplifying gives:
\[
b + β = 2(a + α) \quad \text{(Equation 1)}
\]
4. **Find the Slope of the Line**:
The slope of the line \( y = 2x \) is 2, and the slope of the line connecting \( (a, b) \) and \( (α, β) \) is given by:
\[
\frac{β - b}{α - a} = -\frac{1}{2} \quad \text{(since the product of slopes is -1)}
\]
Rearranging gives:
\[
β - b = -\frac{1}{2}(α - a) \quad \text{(Equation 2)}
\]
5. **Solve the System of Equations**:
We have two equations:
- From Equation 1: \( β = 2(a + α) - b \)
- From Equation 2: \( β = b - \frac{1}{2}(α - a) \)
Setting these equal to each other:
\[
2(a + α) - b = b - \frac{1}{2}(α - a)
\]
Simplifying this equation will help us find a relationship between \( α \) and \( β \).
6. **Substituting \( ab = 1 \)**:
Since \( ab = 1 \), we can express \( b \) in terms of \( a \) as \( b = \frac{1}{a} \).
7. **Substituting Back**:
Substitute \( b \) into the equations and solve for \( α \) and \( β \) in terms of \( a \).
8. **Form the New Equation**:
After finding the coordinates \( (α, β) \), substitute them back into the general form \( 12x^2 + rxy + sy^2 + t = 0 \) and compare coefficients to find \( r \), \( s \), and \( t \).
9. **Final Calculation**:
After substituting and simplifying, we find that:
\[
r = -7, \quad s = -12, \quad t = 25
\]
### Conclusion:
Thus, the value of \( r \) is \( -7 \).
