Let the lines `l_(1)` and `l_(2)` be normals to `y^(2)=4x` and tangents to `x^(2)=-12y` (where `l_(1) and l_(2)` are not x - axis). The absolute value of the difference of slopes of `l_(1) and l_(2)` is

Options

A. 3
B. 2
C. 1
D. `(1)/(2)`

Answer

To solve the problem, we need to find the absolute value of the difference of slopes of lines \( l_1 \) and \( l_2 \) that are normals to the parabola \( y^2 = 4x \) and tangents to the parabola \( x^2 = -12y \).

### Step-by-Step Solution:

1. **Identify the equations of the parabolas:**
- The first parabola is \( y^2 = 4x \).
- The second parabola is \( x^2 = -12y \).

2. **Find the equation of the normal to the first parabola:**
- For the parabola \( y^2 = 4x \), the slope of the normal at a point where the slope of the tangent is \( m \) is given by:
\[ y = mx - 2am - am^3 \]
- Here, \( a = 1 \) (since \( 4a = 4 \)), so the equation becomes:
\[ y = mx - 2m - m^3 \]

3. **Find the equation of the tangent to the second parabola:**
- For the parabola \( x^2 = -12y \), the slope of the tangent at a point where the slope is \( k \) is given by:
\[ y = mx - \frac{m^2}{3} \]
- Here, \( a = -3 \) (since \( -12 = 4a \)), so the equation becomes:
\[ y = kx - 3k^2 \]

4. **Set the equations equal to find the slopes:**
- Since \( l_1 \) is normal to the first parabola and tangent to the second, we set the two equations equal:
\[ mx - 2m - m^3 = kx - 3k^2 \]
- Rearranging gives:
\[ (m - k)x + (3k^2 - 2m - m^3) = 0 \]
- For this to hold for all \( x \), both coefficients must be zero:
1. \( m - k = 0 \) (which gives \( m = k \))
2. \( 3k^2 - 2m - m^3 = 0 \)

5. **Substitute \( m = k \) into the second equation:**
- Substituting \( k \) for \( m \) in the second equation:
\[ 3m^2 - 2m - m^3 = 0 \]
- Rearranging gives:
\[ m^3 - 3m^2 + 2m = 0 \]
- Factoring out \( m \):
\[ m(m^2 - 3m + 2) = 0 \]
- This gives \( m = 0 \) or \( m^2 - 3m + 2 = 0 \).

6. **Solve the quadratic equation:**
- The quadratic \( m^2 - 3m + 2 = 0 \) can be factored as:
\[ (m - 1)(m - 2) = 0 \]
- Thus, \( m = 1 \) or \( m = 2 \).

7. **Determine the slopes of \( l_1 \) and \( l_2 \):**
- The possible slopes are \( m = 1 \) and \( m = 2 \).

8. **Calculate the absolute value of the difference of slopes:**
- The absolute value of the difference of the slopes is:
\[ |1 - 2| = 1 \]

### Final Answer:
The absolute value of the difference of slopes of \( l_1 \) and \( l_2 \) is \( \boxed{1} \).