The family of parabolas with a common vertex at the origin whose foci are on the x-axis and the directrices are parallel to the y-axis can have the equation (a being a parameter)

Options

A. `y^2= 3ax`
B. `x^2=4ay`
C. `y^2 = 4a(x + 2)`
D. `(y)^2=4ax`

Answer

To find the equation of the family of parabolas with a common vertex at the origin, where the foci are on the x-axis and the directrices are parallel to the y-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Configuration**:
- The vertex of the parabola is at the origin (0, 0).
- The focus is on the x-axis, which we can denote as (a, 0).
- The directrix is a vertical line parallel to the y-axis, which can be represented as x = -a.

2. **Equation of a Horizontal Parabola**:
- The standard form of a horizontal parabola centered at (h, k) is given by:
\[ (y - k)^2 = 4p(x - h) \]
- In our case, since the vertex is at the origin (0, 0), we have h = 0 and k = 0. Thus, the equation simplifies to:
\[ y^2 = 4px \]
- Here, 'p' represents the distance from the vertex to the focus.

3. **Identify 'p'**:
- The distance from the vertex (0, 0) to the focus (a, 0) is 'a'. Therefore, we can set \( p = a \).

4. **Substituting p into the Equation**:
- Substituting \( p = a \) into the equation of the parabola gives:
\[ y^2 = 4ax \]

5. **Final Equation**:
- The equation of the family of parabolas is:
\[ y^2 = 4ax \]

### Conclusion:
The correct equation for the family of parabolas with the given conditions is \( y^2 = 4ax \).