To find the equation of the parabola with a focus at (3, 4) and a vertex at the focus of the given parabola \( y^2 - 12x - 4y + 4 = 0 \), we will follow these steps:
### Step 1: Rewrite the given parabola equation
We start with the equation of the given parabola:
\[
y^2 - 12x - 4y + 4 = 0
\]
Rearranging this, we get:
\[
y^2 - 4y + 4 = 12x
\]
### Step 2: Complete the square for y
The left side can be factored as:
\[
(y - 2)^2 = 12x
\]
This is in the standard form of a parabola \( (y - k)^2 = 4p(x - h) \), where the vertex is at (h, k).
### Step 3: Identify the vertex and focus of the given parabola
From the equation \( (y - 2)^2 = 12(x) \), we can see that:
- The vertex \((h, k)\) is \( (0, 2) \).
- The value of \(4p\) is \(12\), hence \(p = 3\).
The focus of this parabola is at:
\[
(h + p, k) = (0 + 3, 2) = (3, 2)
\]
### Step 4: Identify the vertex of the new parabola
The vertex of the new parabola is at the focus of the given parabola, which we found to be \( (3, 2) \). The focus of the new parabola is given as \( (3, 4) \).
### Step 5: Determine the orientation of the new parabola
Since both the vertex and focus share the same x-coordinate (3), the new parabola opens vertically. The vertex is at \( (3, 2) \) and the focus is at \( (3, 4) \).
### Step 6: Calculate the value of p for the new parabola
The distance \(p\) from the vertex to the focus is:
\[
p = 4 - 2 = 2
\]
Thus, the equation of the new parabola can be written as:
\[
(y - 2)^2 = 4p(x - 3)
\]
Substituting \(p = 2\):
\[
(y - 2)^2 = 8(x - 3)
\]
### Step 7: Expand the equation
Expanding this gives:
\[
(y - 2)^2 = 8x - 24
\]
Rearranging it, we get:
\[
y^2 - 4y + 4 = 8x - 24
\]
\[
y^2 - 4y - 8x + 28 = 0
\]
### Final Equation
The final equation of the parabola is:
\[
y^2 - 4y - 8x + 28 = 0
\]
