A line `L` passing through the focus of the parabola `y^2=4(x-1)` intersects the parabola at two distinct points. If `m` is the slope of the line `L ,` then `-11` `m in R` (d) none of these

Options

A. `-1ltmlt1`
B. `mlt-1ormlt`
C. `minR`
D. none of these

Explanation

To solve the problem, we need to analyze the given parabola and the line that intersects it. Let's break down the solution step by step.

### Step 1: Identify the focus of the parabola
The given parabola is \( y^2 = 4(x - 1) \). This can be rewritten in the standard form of a parabola:

\[ y^2 = 4p(x - h) \]

Here, \( h = 1 \) and \( 4p = 4 \), so \( p = 1 \). The vertex of the parabola is at \( (h, 0) = (1, 0) \) and the focus is at \( (h + p, 0) = (2, 0) \).

### Step 2: Write the equation of the line
Let the slope of the line \( L \) be \( m \). The equation of the line passing through the focus \( (2, 0) \) can be written as:

\[ y - 0 = m(x - 2) \]

This simplifies to:

\[ y = mx - 2m \]

### Step 3: Set up the intersection of the line and the parabola
To find the points of intersection between the line and the parabola, substitute \( y = mx - 2m \) into the parabola's equation \( y^2 = 4(x - 1) \):

\[ (mx - 2m)^2 = 4(x - 1) \]

### Step 4: Expand and rearrange the equation
Expanding the left side:

\[ m^2x^2 - 4mx + 4m^2 = 4x - 4 \]

Rearranging gives:

\[ m^2x^2 - (4m + 4)x + (4m^2 + 4) = 0 \]

### Step 5: Determine the conditions for two distinct intersection points
For the line to intersect the parabola at two distinct points, the discriminant of the quadratic equation must be greater than zero. The discriminant \( D \) of the quadratic \( ax^2 + bx + c = 0 \) is given by:

\[ D = b^2 - 4ac \]

In our case, \( a = m^2 \), \( b = -(4m + 4) \), and \( c = 4m^2 + 4 \). Thus, the discriminant is:

\[ D = (-(4m + 4))^2 - 4(m^2)(4m^2 + 4) \]

Calculating this gives:

\[ D = (4m + 4)^2 - 16m^2(1 + m^2) \]

### Step 6: Simplify the discriminant
Expanding \( (4m + 4)^2 \):

\[ D = 16m^2 + 32m + 16 - 16m^2 - 16m^2 \]
\[ D = 32m + 16 - 16m^2 \]

### Step 7: Set the discriminant greater than zero
For two distinct points, we need:

\[ 32m + 16 - 16m^2 > 0 \]

Rearranging gives:

\[ -16m^2 + 32m + 16 > 0 \]

Dividing through by -16 (and reversing the inequality):

\[ m^2 - 2m - 1 < 0 \]

### Step 8: Solve the quadratic inequality
Finding the roots of the equation \( m^2 - 2m - 1 = 0 \):

\[ m = \frac{2 \pm \sqrt{(2)^2 - 4(1)(-1)}}{2(1)} = \frac{2 \pm \sqrt{8}}{2} = 1 \pm \sqrt{2} \]

The roots are \( 1 - \sqrt{2} \) and \( 1 + \sqrt{2} \).

### Step 9: Determine the intervals
The quadratic opens upwards, so the solution to the inequality \( m^2 - 2m - 1 < 0 \) is:

\[ 1 - \sqrt{2} < m < 1 + \sqrt{2} \]

### Conclusion
The slope \( m \) must lie between \( 1 - \sqrt{2} \) and \( 1 + \sqrt{2} \). Since \( 1 - \sqrt{2} < -1 \) and \( 1 + \sqrt{2} > 1 \), we conclude that:

- \( -1 < m < 1 \) is valid.
- \( m < -1 \) or \( m > 1 \) is not valid.

Thus, the answer is **(a) \( -1 < m < 1 \)**.