The vertex and the focus of a parabola are at distances `a` and `a'` respectively form the origin on positive `x-`axis. The equation of the parabola is

Options

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

Explanation

To find the equation of the parabola given the vertex and focus, we can follow these steps:

### Step 1: Identify the coordinates of the vertex and focus
The vertex of the parabola is at the point \( V(a, 0) \) and the focus is at the point \( F(a', 0) \).

### Step 2: Determine the distance between the vertex and the focus
The distance between the vertex and the focus is given by:
\[ d = a' - a \]
This distance is positive if \( a' > a \) and negative if \( a' < a \).

### Step 3: Write the standard form of the parabola
The standard form of a parabola that opens to the left or right with vertex \( (h, k) \) is given by:
\[ (y - k)^2 = 4p(x - h) \]
where \( p \) is the distance from the vertex to the focus. Since our vertex is at \( (a, 0) \), we have \( h = a \) and \( k = 0 \).

### Step 4: Substitute the values into the standard form
Here, \( p = a' - a \). Therefore, the equation becomes:
\[ (y - 0)^2 = 4(a' - a)(x - a) \]
which simplifies to:
\[ y^2 = 4(a' - a)(x - a) \]

### Step 5: Finalize the equation
The final equation of the parabola is:
\[ y^2 = 4(a' - a)(x - a) \]

### Summary
Thus, the equation of the parabola with vertex at \( (a, 0) \) and focus at \( (a', 0) \) is:
\[ y^2 = 4(a' - a)(x - a) \]

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