What is the equation of the parabola, whose vertex and focus are on the `x -` axis at distance a and b from the origin respectively? `(b gt a gt 0)`

Options

A. `y^2 = 8 (b-a) (x-a)`
B. `y^2 = 4(b+a) (x-a)`
C. `y^2 = 4(b-a) (x-a)`
D. `y^2 = 4(b-a) (x-a)`

Answer

To find the equation of the parabola whose vertex and focus are located on the x-axis at distances \(a\) and \(b\) from the origin respectively, we can follow these steps:

### Step 1: Identify the Vertex and Focus
The vertex of the parabola is given at the point \((a, 0)\) and the focus is at the point \((b, 0)\).

### Step 2: Determine the Distance Between Vertex and Focus
The distance between the vertex and the focus is given by:
\[ p = b - a \]
where \(p\) is the distance from the vertex to the focus.

### Step 3: Write the Standard Form of the Parabola
For a parabola that opens to the right, the standard form of the equation is:
\[ y - k = \frac{1}{4p}(x - h)^2 \]
where \((h, k)\) is the vertex. In our case, the vertex is \((a, 0)\), so \(h = a\) and \(k = 0\).

### Step 4: Substitute the Values into the Equation
Substituting \(h\), \(k\), and \(p\) into the standard form gives:
\[ y - 0 = \frac{1}{4(b - a)}(x - a)^2 \]
This simplifies to:
\[ y = \frac{1}{4(b - a)}(x - a)^2 \]

### Step 5: Final Equation of the Parabola
Thus, the equation of the parabola is:
\[ y = \frac{1}{4(b - a)}(x - a)^2 \]