Points `A and B` lie on the parabola `y = 2x^2 + 4x - 2,` such that origin is the mid-point of the linesegment `AB.` If `l` be the length of the line segment `AB,` then find the unit digit of `l^2.`

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript while providing a clearer structure.

### Step 1: Identify the Parabola
The equation of the parabola is given as:
\[ y = 2x^2 + 4x - 2 \]

### Step 2: Find the Vertex of the Parabola
To rewrite the parabola in vertex form, we can complete the square:
1. Factor out the coefficient of \(x^2\):
\[ y = 2(x^2 + 2x) - 2 \]
2. Complete the square inside the parentheses:
\[ y = 2\left((x + 1)^2 - 1\right) - 2 \]
\[ y = 2(x + 1)^2 - 2 - 2 \]
\[ y = 2(x + 1)^2 - 4 \]

The vertex of the parabola is at the point \((-1, -4)\).

### Step 3: Determine the Points A and B
Given that the origin (0, 0) is the midpoint of the line segment AB, we can denote the coordinates of points A and B as:
- \( A(x_1, y_1) \)
- \( B(x_2, y_2) \)

Since the midpoint is at the origin:
\[ \frac{x_1 + x_2}{2} = 0 \Rightarrow x_1 + x_2 = 0 \]
This implies \( x_2 = -x_1 \).

### Step 4: Find the Equation of Line AB
The line segment AB can be represented as:
\[ y = mx \]
Since it passes through the origin.

### Step 5: Find the Intersection of Line AB with the Parabola
Substituting \(y = mx\) into the parabola's equation:
\[ mx = 2x^2 + 4x - 2 \]
Rearranging gives:
\[ 2x^2 + (4 - m)x - 2 = 0 \]

### Step 6: Use the Condition for the Midpoint
From the quadratic equation, the sum of the roots \(x_1 + x_2\) can be expressed as:
\[ x_1 + x_2 = -\frac{b}{a} = -\frac{4 - m}{2} \]
Setting this equal to zero (from the midpoint condition):
\[ -\frac{4 - m}{2} = 0 \]
This leads to:
\[ 4 - m = 0 \Rightarrow m = 4 \]

### Step 7: Substitute \(m\) Back into the Line Equation
Now we have the equation of line AB:
\[ y = 4x \]

### Step 8: Find the Points of Intersection
Substituting \(y = 4x\) back into the parabola:
\[ 4x = 2x^2 + 4x - 2 \]
This simplifies to:
\[ 2x^2 - 2 = 0 \]
\[ x^2 = 1 \Rightarrow x = 1 \text{ or } x = -1 \]

### Step 9: Calculate the Corresponding y-coordinates
For \(x = 1\):
\[ y = 4(1) = 4 \Rightarrow A(1, 4) \]
For \(x = -1\):
\[ y = 4(-1) = -4 \Rightarrow B(-1, -4) \]

### Step 10: Calculate the Length of Segment AB
The length \(l\) of segment AB is given by the distance formula:
\[ l = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of A and B:
\[ l = \sqrt{((-1) - 1)^2 + ((-4) - 4)^2} = \sqrt{(-2)^2 + (-8)^2} = \sqrt{4 + 64} = \sqrt{68} \]

### Step 11: Find \(l^2\) and its Unit Digit
Calculating \(l^2\):
\[ l^2 = 68 \]
The unit digit of \(l^2\) is \(8\).

### Final Answer
The unit digit of \(l^2\) is \(8\).