The slope of line which belongs to family (1+ l) x + (l-1)y + 2(1-l) = 0 and makes shortest intercept on `x^2 = 4y - 4`

Explanation

To solve the problem, we need to find the slope of the line from the given family that makes the shortest intercept on the parabola defined by \( x^2 = 4y - 4 \).

### Step-by-Step Solution:

1. **Identify the Family of Lines**:
The family of lines is given by the equation:
\[ (1 + l)x + (l - 1)y + 2(1 - l) = 0 \]
Rearranging this, we can express it in the standard form \( Ax + By + C = 0 \).

2. **Rearranging the Equation**:
Expanding and rearranging gives:
\[ (1 + l)x + (l - 1)y + 2 - 2l = 0 \]
Which can be rewritten as:
\[ (1 + l)x + (l - 1)y = 2l - 2 \]

3. **Finding the Slope**:
The slope \( m \) of the line can be found from the equation:
\[ y = -\frac{(1 + l)}{(l - 1)}x + \frac{(2l - 2)}{(l - 1)} \]
Thus, the slope \( m \) is:
\[ m = -\frac{(1 + l)}{(l - 1)} \]

4. **Equation of the Parabola**:
The parabola is given by:
\[ x^2 = 4y - 4 \quad \Rightarrow \quad y = \frac{x^2}{4} + 1 \]

5. **Finding the Intercept**:
The intercept of the line on the y-axis occurs when \( x = 0 \):
\[ y = \frac{(2l - 2)}{(l - 1)} \]
The x-intercept occurs when \( y = 0 \):
\[ 0 = -\frac{(1 + l)}{(l - 1)}x + \frac{(2l - 2)}{(l - 1)} \quad \Rightarrow \quad x = \frac{2l - 2}{(1 + l)} \]

6. **Calculating the Length of the Intercept**:
The length of the intercept can be calculated as:
\[ \text{Length} = \left| \frac{2l - 2}{(1 + l)} \right| + \left| \frac{(2l - 2)}{(l - 1)} \right| \]

7. **Minimizing the Intercept**:
To minimize the intercept, we can differentiate the intercept length with respect to \( l \) and set the derivative to zero. However, since we are looking for the shortest intercept, we can also analyze the conditions under which the intercept is minimized.

8. **Finding the Slope for Shortest Intercept**:
The shortest intercept occurs when the line is vertical or horizontal. In this case, the line that makes the shortest intercept with the parabola is horizontal, which means the slope \( m = 0 \).

### Final Answer:
The slope of the line which belongs to the family and makes the shortest intercept on the parabola is:
\[ \boxed{0} \]