`lx +my = 1` is the equation of the chord PQ of `y^(2) = 4x` whose focus is S. If PS and QS meet the parabola again at R and T respectively, then slope of RT is

Options

A. `-(1)/(m)`
B. `(l)/(m)`
C. `(2)/(m)`
D. none of these

Explanation

To solve the problem step by step, we will analyze the given information and derive the required slope of the line segment RT.

### Step 1: Understand the Parabola and Focus
The equation of the parabola is given as \( y^2 = 4x \). The focus \( S \) of this parabola is at the point \( (1, 0) \).

**Hint:** Recall that the focus of the parabola \( y^2 = 4px \) is at \( (p, 0) \).

### Step 2: Identify Points P and Q
The chord \( PQ \) is given by the equation \( Lx + My = 1 \). We can express the points \( P \) and \( Q \) on the parabola in terms of parameters \( t_1 \) and \( t_2 \):
- \( P(t_1) = (t_1^2, 2t_1) \)
- \( Q(t_2) = (t_2^2, 2t_2) \)

**Hint:** Use the parametric form of the parabola to identify points.

### Step 3: Find the Coordinates of Points R and T
The points \( R \) and \( T \) are where the lines \( PS \) and \( QS \) intersect the parabola again. The coordinates of \( R \) and \( T \) can be expressed as:
- \( R = \left( \frac{1}{t_1^2}, -\frac{2}{t_1} \right) \)
- \( T = \left( \frac{1}{t_2^2}, -\frac{2}{t_2} \right) \)

**Hint:** Use the properties of the parabola to find these intersection points.

### Step 4: Calculate the Slope of Line RT
The slope \( m \) of the line segment \( RT \) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of \( R \) and \( T \):
\[ m = \frac{-\frac{2}{t_2} + \frac{2}{t_1}}{\frac{1}{t_2^2} - \frac{1}{t_1^2}} \]

**Hint:** Simplify the expression for the slope step by step.

### Step 5: Simplify the Slope Expression
The expression can be simplified as follows:
\[ m = \frac{2 \left( \frac{1}{t_1} - \frac{1}{t_2} \right)}{\frac{1}{t_2^2} - \frac{1}{t_1^2}} = \frac{2 \left( \frac{t_2 - t_1}{t_1 t_2} \right)}{\frac{t_1^2 - t_2^2}{t_1^2 t_2^2}} = \frac{2(t_2 - t_1) t_1^2 t_2^2}{(t_1 - t_2)(t_1 + t_2)} \]
This simplifies to:
\[ m = \frac{2t_1 t_2}{t_1 + t_2} \]

**Hint:** Factor and cancel terms carefully.

### Step 6: Relate Slope to Given Parameters
Using the relationships \( t_1 + t_2 = -\frac{2M}{L} \) and \( t_1 t_2 = -\frac{1}{L} \), we can substitute these into the slope:
\[ m = \frac{2 \left(-\frac{1}{L}\right)}{-\frac{2M}{L}} = \frac{1}{M} \]

**Hint:** Substitute the relationships derived from the chord equation.

### Conclusion
The slope of the line segment \( RT \) is given by:
\[ \text{slope of } RT = \frac{1}{M} \]

This concludes the solution to the problem.