Find the equation of the parabola whose focus is S (1,-7) and vertex is A(1,-2).

Answer

To find the equation of the parabola with a focus at S(1, -7) and a vertex at A(1, -2), we can follow these steps:

### Step 1: Identify the vertex and focus
The vertex A is given as (1, -2) and the focus S is given as (1, -7).

### Step 2: Determine the orientation of the parabola
Since both the vertex and focus have the same x-coordinate (1), the parabola opens vertically. The vertex is above the focus, indicating that the parabola opens downwards.

### Step 3: Use the standard form of the parabola's equation
The standard form of the equation of a parabola that opens vertically is:
\[ (x - h)^2 = 4a(y - k) \]
where (h, k) is the vertex.

### Step 4: Identify h and k
From the vertex A(1, -2), we have:
- \(h = 1\)
- \(k = -2\)

### Step 5: Calculate the value of 'a'
The distance 'a' is the distance from the vertex to the focus. Since the vertex is at (1, -2) and the focus is at (1, -7), we can calculate 'a':
\[ a = k - \text{(y-coordinate of focus)} = -2 - (-7) = -2 + 7 = 5 \]
Since the parabola opens downwards, 'a' will be negative:
\[ a = -5 \]

### Step 6: Substitute h, k, and a into the standard equation
Now we can substitute \(h\), \(k\), and \(a\) into the standard form:
\[ (x - 1)^2 = 4(-5)(y + 2) \]

### Step 7: Simplify the equation
This simplifies to:
\[ (x - 1)^2 = -20(y + 2) \]

### Final Equation
Thus, the equation of the parabola is:
\[ (x - 1)^2 = -20(y + 2) \]

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