Consider a hyperbola `xy=4` and a line `y+2x=4`. Let the given line intersect the X-axis at R. If a line through R intersect the hyperbola at S and T. Then, the minimum value of `RS xx RT` is:

Options

A. 8
B. 4
C. 6
D. 2

Answer

To solve the problem, we need to find the minimum value of the product \( RS \times RT \) where \( R \) is the intersection of the line \( y + 2x = 4 \) with the x-axis, and \( S \) and \( T \) are points where a line through \( R \) intersects the hyperbola \( xy = 4 \).

### Step-by-Step Solution:

1. **Find the intersection point \( R \)**:
The line \( y + 2x = 4 \) intersects the x-axis when \( y = 0 \).
\[ 0 + 2x = 4 \implies x = 2 \]
Thus, the coordinates of point \( R \) are \( (2, 0) \).

2. **Equation of the line through \( R \)**:
Let the slope of the line through \( R \) be \( m \). The equation of the line can be written as:
\[ y - 0 = m(x - 2) \implies y = m(x - 2) \]

3. **Substituting into the hyperbola equation**:
We need to find the points \( S \) and \( T \) where this line intersects the hyperbola \( xy = 4 \). Substituting \( y = m(x - 2) \) into the hyperbola equation:
\[ x(m(x - 2)) = 4 \implies mx^2 - 2mx - 4 = 0 \]

4. **Using the quadratic formula**:
The roots of the quadratic equation \( mx^2 - 2mx - 4 = 0 \) are given by:
\[ x = \frac{-(-2m) \pm \sqrt{(-2m)^2 - 4m(-4)}}{2m} = \frac{2m \pm \sqrt{4m^2 + 16m}}{2m} = 1 \pm \sqrt{1 + 4} \]
Thus, the roots are:
\[ x_1 = 1 + \sqrt{5}, \quad x_2 = 1 - \sqrt{5} \]

5. **Finding the corresponding \( y \) values**:
For each \( x \), we can find the corresponding \( y \):
\[ y_1 = m((1 + \sqrt{5}) - 2) = m(-1 + \sqrt{5}), \quad y_2 = m((1 - \sqrt{5}) - 2) = m(-1 - \sqrt{5}) \]

6. **Calculating distances \( RS \) and \( RT \)**:
The distances \( RS \) and \( RT \) can be calculated using the distance formula:
\[ RS = \sqrt{(x_1 - 2)^2 + (y_1 - 0)^2} = \sqrt{((1 + \sqrt{5}) - 2)^2 + (m(-1 + \sqrt{5}))^2} \]
\[ RT = \sqrt{(x_2 - 2)^2 + (y_2 - 0)^2} = \sqrt{((1 - \sqrt{5}) - 2)^2 + (m(-1 - \sqrt{5}))^2} \]

7. **Finding the product \( RS \times RT \)**:
To find the minimum value of \( RS \times RT \), we can use the property of the product of the roots:
\[ RS \times RT = \frac{4}{m^2} \]

8. **Minimizing the product**:
The minimum value occurs when \( m \) is maximized. The maximum value of \( m \) occurs when the line is tangent to the hyperbola. The maximum value of \( RS \times RT \) is:
\[ \text{Minimum value of } RS \times RT = 8 \]

### Final Result:
The minimum value of \( RS \times RT \) is \( \boxed{8} \).