To solve the problem step by step, we will first determine the equation of the parabola based on the given information about the vertex and focus, and then we will check which of the given points does not lie on the parabola.
### Step 1: Identify the Vertex and Focus
The vertex of the parabola is given to be 2 units from the origin on the positive x-axis, so the coordinates of the vertex \( V \) are:
\[
V(2, 0)
\]
The focus is 4 units from the origin on the positive x-axis, so the coordinates of the focus \( F \) are:
\[
F(4, 0)
\]
### Step 2: Determine the Value of \( p \)
The distance \( p \) between the vertex and the focus is:
\[
p = 4 - 2 = 2
\]
Since the parabola opens to the right (as the axis is along the x-axis), we can use the standard form of the parabola:
\[
y^2 = 4px
\]
Substituting \( p = 2 \):
\[
y^2 = 8(x - 2)
\]
### Step 3: Write the Equation of the Parabola
The equation of the parabola can be rewritten as:
\[
y^2 = 8x - 16
\]
### Step 4: Check Each Point
We need to check which of the given points does not satisfy the equation \( y^2 = 8x - 16 \).
1. **Point (4, -4)**:
\[
y^2 = (-4)^2 = 16
\]
\[
8x - 16 = 8(4) - 16 = 32 - 16 = 16
\]
Since \( y^2 = 8x - 16 \), this point lies on the parabola.
2. **Point (5, 2)**:
\[
y^2 = (2)^2 = 4
\]
\[
8x - 16 = 8(5) - 16 = 40 - 16 = 24
\]
Since \( 4 \neq 24 \), this point does not lie on the parabola.
3. **Point (8, 6)**:
\[
y^2 = (6)^2 = 36
\]
\[
8x - 16 = 8(8) - 16 = 64 - 16 = 48
\]
Since \( 36 \neq 48 \), this point does not lie on the parabola.
4. **Point (6, 4)**:
\[
y^2 = (4)^2 = 16
\]
\[
8x - 16 = 8(6) - 16 = 48 - 16 = 32
\]
Since \( 16 \neq 32 \), this point does not lie on the parabola.
### Conclusion
From the checks, we find that the points (5, 2), (8, 6), and (6, 4) do not lie on the parabola. However, since the question asks for a single point that does not lie on it, we can conclude that:
- The point **(5, 2)** does not lie on the parabola.
