A line L passing through the focus of the parabola `(y-2)^(2)=4(x+1)` intersects the two distinct point. If m be the slope of the line I,, then

Options

A. `min(-oo, -1)uu(1, oo)`
B. `m in (-oo, 0)uu(0, oo)`
C. `min(-oo, 0)uu(0, oo)`
D. none of these

Answer

To solve the problem, we need to analyze the parabola given by the equation \((y - 2)^2 = 4(x + 1)\) and determine the range of slopes \(m\) for a line passing through the focus of the parabola that intersects it at two distinct points.

### Step-by-Step Solution:

1. **Identify the Standard Form of the Parabola**:
The given equation \((y - 2)^2 = 4(x + 1)\) is in the standard form of a parabola that opens to the right. The standard form is \((y - k)^2 = 4p(x - h)\), where \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus.

2. **Find the Vertex and Focus**:
From the equation, we can identify:
- Vertex \((h, k) = (-1, 2)\)
- The value of \(4p = 4\), thus \(p = 1\).
- The focus is located at \((h + p, k) = (-1 + 1, 2) = (0, 2)\).

3. **Equation of the Line**:
A line passing through the focus \((0, 2)\) can be expressed in slope-intercept form as:
\[ y - 2 = m(x - 0) \implies y = mx + 2 \]

4. **Set Up the Intersection**:
To find the points of intersection between the line and the parabola, we substitute \(y = mx + 2\) into the parabola's equation:
\[ (mx + 2 - 2)^2 = 4(x + 1) \]
Simplifying this gives:
\[ (mx)^2 = 4(x + 1) \implies m^2x^2 = 4x + 4 \]
Rearranging leads to:
\[ m^2x^2 - 4x - 4 = 0 \]

5. **Determine Conditions for Two Distinct Intersections**:
For the quadratic equation \(m^2x^2 - 4x - 4 = 0\) to have two distinct solutions, the discriminant must be greater than zero:
\[ D = b^2 - 4ac = (-4)^2 - 4(m^2)(-4) = 16 + 16m^2 \]
Setting the discriminant greater than zero:
\[ 16 + 16m^2 > 0 \]
This inequality is always true for all real values of \(m\).

6. **Special Case for Horizontal Line**:
The only exception occurs when \(m = 0\) (the horizontal line \(y = 2\)), which intersects the parabola at exactly one point. Therefore, \(m = 0\) must be excluded from the range.

7. **Conclusion**:
The range of slopes \(m\) for which the line intersects the parabola at two distinct points is:
\[ m \in (-\infty, 0) \cup (0, \infty) \]