Axis of a parabola lies along x-axis. If its vertex and focus are at distances 2 and 4, respectively, from the origin on the positive x-axis, then which of the following points does not lie on it ?

Options

A. (8,6)
B. (4,-4)
C. `(6,4sqrt(2))`
D. `(5,2sqrt(6))`

Explanation

To solve the problem, we need to find the equation of the parabola and then check which of the given points does not lie on it. Here’s a step-by-step solution:

### Step 1: Identify the vertex and focus of the parabola
The vertex of the parabola is given to be at a distance of 2 from the origin along the positive x-axis. Therefore, the vertex is at the point \( (2, 0) \).

The focus is at a distance of 4 from the origin along the positive x-axis, so the focus is at the point \( (4, 0) \).

### Step 2: Determine the value of \( a \)
In a parabola, the distance between the vertex and the focus is denoted as \( a \). Here, the distance from the vertex \( (2, 0) \) to the focus \( (4, 0) \) is:
\[ a = 4 - 2 = 2 \]

### Step 3: Write the equation of the parabola
Since the axis of the parabola is along the x-axis and the vertex is at \( (h, k) = (2, 0) \), the standard form of the equation of the parabola can be written as:
\[ y^2 = 4a(x - h) \]
Substituting \( a = 2 \) and \( h = 2 \):
\[ y^2 = 4 \cdot 2 \cdot (x - 2) \]
This simplifies to:
\[ y^2 = 8(x - 2) \]

### Step 4: Check the given points
Now, we need to check which of the given points does not lie on this parabola. Let's assume we have a point \( (x_1, y_1) \) to test.

1. **For point \( (8, 6) \)**:
\[ y^2 = 6^2 = 36 \]
\[ 8(x - 2) = 8(8 - 2) = 8 \cdot 6 = 48 \]
Since \( 36 \neq 48 \), the point \( (8, 6) \) does not lie on the parabola.

2. **For other points**, you would substitute their coordinates into the equation \( y^2 = 8(x - 2) \) and check if both sides are equal.

### Conclusion
The point that does not lie on the parabola is \( (8, 6) \).