A parabola has the origin as its focus and the line x = 2 as the directrix. Then the vertex of the parabola is at

Options

A. (2,0)
B. (0,2)
C. (1,0)
D. `(0,1)`

Answer

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

### Step 2: Understand the Definition of a Parabola
A parabola is defined as the set of all points \( P(x, y) \) that are equidistant from the focus and the directrix. Therefore, for any point \( P \) on the parabola, the distance from \( P \) to the focus \( F \) must equal the distance from \( P \) to the directrix.

### Step 3: Set Up the Distance Equations
Let \( P(x, y) \) be a point on the parabola. The distance from \( P \) to the focus \( F(0, 0) \) is:
\[ d_1 = \sqrt{(x - 0)^2 + (y - 0)^2} = \sqrt{x^2 + y^2} \]

The distance from \( P \) to the directrix \( x = 2 \) is:
\[ d_2 = |x - 2| \]

### Step 4: Set the Distances Equal
According to the definition of the parabola:
\[ \sqrt{x^2 + y^2} = |x - 2| \]

### Step 5: Square Both Sides
To eliminate the square root, we square both sides:
\[ x^2 + y^2 = (x - 2)^2 \]

### Step 6: Expand and Rearrange
Expanding the right side:
\[ x^2 + y^2 = x^2 - 4x + 4 \]
Now, subtract \( x^2 \) from both sides:
\[ y^2 = -4x + 4 \]

### Step 7: Identify the Vertex
The vertex of the parabola can be found by rewriting the equation:
\[ y^2 = -4(x - 1) \]
This shows that the vertex is at the point \( (1, 0) \).

### Conclusion
Thus, the vertex of the parabola is at the point \( (1, 0) \).