Find the equation of the parabola whose focus is (4,-3) and vertex is (4,-1).

Answer

To find the equation of the parabola with the given focus and vertex, we can follow these steps:

### Step 1: Identify the given points
The focus is given as \( F(4, -3) \) and the vertex is given as \( V(4, -1) \).

### Step 2: Determine the orientation of the parabola
Since both the focus and vertex have the same x-coordinate (4), the axis of the parabola is vertical. The vertex is above the focus, indicating that the parabola opens downwards.

### Step 3: Use the standard form of the parabola
The standard form of the equation of a parabola that opens downwards is:
\[ (x - h)^2 = -4a(y - k) \]
where \( (h, k) \) is the vertex and \( a \) is the distance from the vertex to the focus.

### Step 4: Identify the vertex coordinates
From the vertex \( V(4, -1) \), we have:
- \( h = 4 \)
- \( k = -1 \)

### Step 5: Calculate the value of \( a \)
The distance \( a \) can be calculated as the distance from the vertex to the focus. The y-coordinates of the focus and vertex are:
- Vertex y-coordinate: \( -1 \)
- Focus y-coordinate: \( -3 \)

Thus, the distance \( a \) is:
\[ a = |(-1) - (-3)| = |-1 + 3| = |2| = 2 \]

### Step 6: Substitute values into the standard form
Now substituting \( h = 4 \), \( k = -1 \), and \( a = 2 \) into the standard form:
\[ (x - 4)^2 = -4(2)(y + 1) \]
This simplifies to:
\[ (x - 4)^2 = -8(y + 1) \]

### Step 7: Rearrange to standard form
Expanding and rearranging gives:
\[ (x - 4)^2 = -8y - 8 \]
\[ x^2 - 8x + 16 = -8y - 8 \]
\[ x^2 - 8x + 8y + 24 = 0 \]

### Final Equation
Thus, the equation of the parabola is:
\[ x^2 - 8x + 8y + 24 = 0 \]

---