If axis of the parabola lie along y-axis and if vertex is at a distance of 2 from origin on positive y-axis, focus is at a distance of 2 from origin on negative y-axis then which of the following points doesn't lie on the parabola:

Options

A. `(4sqrt(2), 0)`
B. `(4,1)`
C. `(-4, -2)`
D. `(8, - 2)`

Explanation

To solve the problem, we need to determine the equation of the parabola based on the given information and then check which of the provided points does not lie on the parabola.

### Step 1: Understand the Parabola's Properties
Given:
- The axis of the parabola lies along the y-axis.
- The vertex is at a distance of 2 from the origin on the positive y-axis, which means the vertex is at (0, 2).
- The focus is at a distance of 2 from the origin on the negative y-axis, which means the focus is at (0, -2).

### Step 2: Determine the Value of 'a'
The distance between the vertex and the focus is denoted by 'a'. Here, the vertex is at (0, 2) and the focus is at (0, -2). Thus:
- The distance 'a' = |2 - (-2)| = 4.

### Step 3: Write the Equation of the Parabola
The standard form of a parabola that opens downwards with vertex at (h, k) is given by:
\[ (x - h)^2 = -4a(y - k) \]
Substituting the values:
- \(h = 0\), \(k = 2\), and \(a = 4\):
\[ x^2 = -16(y - 2) \]
This simplifies to:
\[ x^2 = -16y + 32 \quad \text{or} \quad x^2 + 16y - 32 = 0 \]

### Step 4: Check Each Point
Now we will check each of the provided points to see if they satisfy the equation of the parabola.

1. **Point (4√2, 0)**:
\[ (4\sqrt{2})^2 + 16(0) - 32 = 32 - 32 = 0 \quad \text{(lies on the parabola)} \]

2. **Point (4, 1)**:
\[ (4)^2 + 16(1) - 32 = 16 + 16 - 32 = 0 \quad \text{(lies on the parabola)} \]

3. **Point (-4, -2)**:
\[ (-4)^2 + 16(-2) - 32 = 16 - 32 - 32 = -48 \quad \text{(does not lie on the parabola)} \]

### Conclusion
The point that does not lie on the parabola is **(-4, -2)**.