Find the equation of parabola whose focus is (0,1) and the directrix is x+2=0. Also find the vertex of the parabola.

Explanation

To find the equation of the parabola with a focus at (0, 1) and a directrix given by the line x + 2 = 0 (or x = -2), we can follow these steps:

### Step 1: Identify the focus and directrix
- The focus of the parabola is given as \( F(0, 1) \).
- The directrix is \( x + 2 = 0 \), which simplifies to \( x = -2 \).

### Step 2: Determine the vertex
- The vertex (V) of the parabola lies on the axis of symmetry, which is the line that is perpendicular to the directrix and passes through the focus.
- The directrix is a vertical line (x = -2), so the axis of symmetry is a horizontal line through the focus at \( y = 1 \).
- The x-coordinate of the vertex is the midpoint between the focus and the directrix.

To find the x-coordinate of the vertex:
- The x-coordinate of the focus is 0, and the x-coordinate of the directrix is -2.
- The midpoint (x-coordinate) is calculated as:
\[ x_v = \frac{0 + (-2)}{2} = \frac{-2}{2} = -1 \]
- Therefore, the vertex \( V \) is at \( (-1, 1) \).

### Step 3: Determine the value of \( a \)
- The distance \( a \) is the distance from the vertex to the focus (or from the vertex to the directrix).
- The distance from the vertex \( (-1, 1) \) to the focus \( (0, 1) \) is:
\[ a = |0 - (-1)| = 1 \]

### Step 4: Write the equation of the parabola
- Since the parabola opens to the right (the focus is to the right of the directrix), the standard form of the equation of the parabola is:
\[ (y - k)^2 = 4a(x - h) \]
where \( (h, k) \) is the vertex.
- Substituting \( h = -1 \), \( k = 1 \), and \( a = 1 \):
\[ (y - 1)^2 = 4 \cdot 1 \cdot (x + 1) \]
Simplifying this gives:
\[ (y - 1)^2 = 4(x + 1) \]

### Step 5: Conclusion
- The equation of the parabola is:
\[ (y - 1)^2 = 4(x + 1) \]
- The vertex of the parabola is at \( (-1, 1) \).