If a parabola has the origin as its focus and the line `x = 2` as the directrix, then the coordinates of the vertex of the parabola are

Options

A. (0,1)
B. (1,0)
C. (0,-1)
D. (-1,0)

Answer

To find the coordinates of the vertex of the parabola with the origin as its focus and the line \( x = 2 \) as its directrix, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Focus and Directrix**:
- The focus \( S \) of the parabola is given as the origin, which is \( (0, 0) \).
- The directrix is given as the line \( x = 2 \).

2. **Determine the Foot of the Directrix**:
- The foot of the directrix is the point on the directrix that is closest to the focus. Since the directrix is a vertical line \( x = 2 \), the foot of the directrix \( D \) is at the point \( (2, 0) \).

3. **Find the Vertex**:
- The vertex \( V \) of the parabola is the midpoint between the focus \( S \) and the foot of the directrix \( D \).
- To find the midpoint, we use the midpoint formula:
\[ V = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
- Here, \( S = (0, 0) \) and \( D = (2, 0) \).
- Plugging in the coordinates:
\[ V = \left( \frac{0 + 2}{2}, \frac{0 + 0}{2} \right) = \left( \frac{2}{2}, 0 \right) = (1, 0) \]

4. **Conclusion**:
- Therefore, the coordinates of the vertex of the parabola are \( (1, 0) \).

### Final Answer:
The coordinates of the vertex of the parabola are \( (1, 0) \).