Find the equation of the parabola whose vertex and focus are on the positive X-axis at a distance of a and a' from the origin respectively.

Answer

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

### Step-by-Step Solution:

1. **Identify the Coordinates of the Vertex and Focus:**
- The vertex of the parabola is located at \( (a, 0) \).
- The focus of the parabola is located at \( (a', 0) \).

2. **Determine the Distance \( p \):**
- The distance \( p \) between the vertex and the focus is given by:
\[ p = a' - a \]
- Here, \( p \) is the distance from the vertex to the focus along the x-axis.

3. **General Form of the Parabola:**
- The standard form of a parabola that opens to the right is:
\[ (y - k)^2 = 4p(x - h) \]
- Where \( (h, k) \) is the vertex of the parabola.

4. **Substituting the Vertex Coordinates:**
- In our case, the vertex \( (h, k) \) is \( (a, 0) \). Thus, substituting these values into the equation gives:
\[ (y - 0)^2 = 4p(x - a) \]
- This simplifies to:
\[ y^2 = 4p(x - a) \]

5. **Substituting the Value of \( p \):**
- We already found that \( p = a' - a \). Substituting this into the equation gives:
\[ y^2 = 4(a' - a)(x - a) \]

6. **Final Equation of the Parabola:**
- Therefore, the equation of the parabola is:
\[ y^2 = 4(a' - a)(x - a) \]