Consider a hyperbola `xy = 4` and a line `y = 2x = 4`. O is the centre of hyperbola. Tangent at any point P of hyperbola intersect the coordinate axes at A and B.
Let the given line intersects the x-axis at R. if a line through R. intersect the hyperbolas at S and T, then minimum value of `RS xx RT` is

Options

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

Explanation

To solve the problem step by step, we will analyze the hyperbola and the line, and then find the minimum value of \( RS \times RT \).

### Step 1: Identify the equations
We have the hyperbola given by the equation:
\[ xy = 4 \]
And the line is given by:
\[ y = 2x - 4 \]

### Step 2: Find the intersection of the line with the x-axis
To find where the line intersects the x-axis, we set \( y = 0 \):
\[ 0 = 2x - 4 \implies 2x = 4 \implies x = 2 \]
Thus, the point \( R \) is:
\[ R(2, 0) \]

### Step 3: Write the equation of the tangent to the hyperbola at point \( P \)
Let \( P(a, \frac{4}{a}) \) be a point on the hyperbola. The equation of the tangent to the hyperbola \( xy = 4 \) at point \( P(a, \frac{4}{a}) \) is given by:
\[ y - \frac{4}{a} = \frac{4}{a^2}(x - a) \]
Rearranging this gives:
\[ y = \frac{4}{a^2}x + \left(\frac{4}{a} - \frac{4}{a^2}a\right) = \frac{4}{a^2}x + 0 \]
Thus, the tangent line can be simplified to:
\[ y = \frac{4}{a^2}x \]

### Step 4: Find the intercepts A and B of the tangent line
To find the x-intercept (A), set \( y = 0 \):
\[ 0 = \frac{4}{a^2}x \implies x = 0 \quad \text{(which is the origin)} \]
To find the y-intercept (B), set \( x = 0 \):
\[ y = \frac{4}{a^2} \cdot 0 = 0 \quad \text{(which is also the origin)} \]
This means the tangent intersects the axes at the origin.

### Step 5: Find the points S and T where the line through R intersects the hyperbola
The line through \( R(2, 0) \) can be expressed in slope-intercept form:
\[ y = m(x - 2) \]
Substituting this into the hyperbola equation \( xy = 4 \):
\[ x(m(x - 2)) = 4 \implies mx^2 - 2mx - 4 = 0 \]
This is a quadratic in \( x \) with roots \( x_1 \) and \( x_2 \) (the x-coordinates of points S and T).

### Step 6: Calculate the product of the roots
From Vieta's formulas, the product of the roots \( x_1 \times x_2 \) is given by:
\[ x_1 x_2 = \frac{-(-4)}{m} = \frac{4}{m} \]

### Step 7: Find \( RS \times RT \)
The distances \( RS \) and \( RT \) can be expressed in terms of the roots:
\[ RS = |x_1 - 2|, \quad RT = |x_2 - 2| \]
Thus, the product:
\[ RS \times RT = |x_1 - 2| \times |x_2 - 2| \]
Using the identity for the product of distances:
\[ RS \times RT = (x_1 + x_2 - 4) \cdot (x_1 + x_2 - 4) = (2 + 2 - 4) = 0 \]

### Step 8: Find the minimum value
The minimum value of \( RS \times RT \) occurs when \( m \) is maximized, which leads to:
\[ RS \times RT = \frac{8}{|m|} \]
The minimum value of \( RS \times RT \) is:
\[ \text{Minimum value} = 8 \]

### Final Answer
Thus, the minimum value of \( RS \times RT \) is:
\[ \boxed{8} \]