Equation of parabola whose vertex is (-1, 2) and focus is (3, 2) is

Answer

To find the equation of the parabola with vertex (-1, 2) and focus (3, 2), we can follow these steps:

### Step 1: Identify the Vertex and Focus
The vertex of the parabola is given as \( V(-1, 2) \) and the focus is given as \( F(3, 2) \).

### Step 2: Determine the Orientation of the Parabola
Since the y-coordinates of the vertex and focus are the same (both are 2), the parabola opens horizontally. Specifically, it opens to the right because the focus is to the right of the vertex.

### Step 3: Calculate the Distance \( a \)
The distance \( a \) between the vertex and the focus can be calculated as follows:
\[ a = \text{Distance from vertex to focus} = x_{\text{focus}} - x_{\text{vertex}} = 3 - (-1) = 3 + 1 = 4 \]

### Step 4: Write the Standard Form of the Parabola
The standard form of a horizontally opening parabola is given by:
\[ (y - y_1)^2 = 4a(x - x_1) \]
where \( (x_1, y_1) \) is the vertex.

### Step 5: Substitute the Values into the Equation
From our vertex \( V(-1, 2) \):
- \( x_1 = -1 \)
- \( y_1 = 2 \)
- \( a = 4 \)

Substituting these values into the standard form:
\[ (y - 2)^2 = 4 \cdot 4 \cdot (x - (-1)) \]
This simplifies to:
\[ (y - 2)^2 = 16(x + 1) \]

### Final Equation
Thus, the equation of the parabola is:
\[ (y - 2)^2 = 16(x + 1) \]

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