A parabola is drawn with focus at (3,4) and vertex at the focus of the parabola `y^2-12x-4y+4=0`. The equation of the parabola is

Options

A. (a)`x^(2)-6x-8y+25=0`
B. (b)`y^(2)-8x-4y+28=0`
C. (c)`x^(2)-6x+8y-25=0`
D. (d)`x^(2)-4x-8y-28=0`

Explanation

To find the equation of the parabola with a focus at (3, 4) and a vertex at the focus of the parabola given by the equation \(y^2 - 12x - 4y + 4 = 0\), we will follow these steps:

### Step 1: Convert the given equation of the parabola into standard form.
The given equation is:
\[ y^2 - 12x - 4y + 4 = 0 \]
Rearranging it, we have:
\[ y^2 - 4y = 12x - 4 \]
Completing the square on the left side:
\[ (y - 2)^2 - 4 = 12x - 4 \]
Adding 4 to both sides gives:
\[ (y - 2)^2 = 12x \]
This can be rewritten as:
\[ (y - 2)^2 = 4(3)(x - 0) \]
From this, we can identify the vertex of the parabola as \((0, 2)\).

### Step 2: Identify the vertex and focus of the new parabola.
We know the focus of the new parabola is given as \((3, 4)\) and the vertex is the focus of the previous parabola, which we found to be \((0, 2)\).

### Step 3: Determine the value of \(p\).
The distance \(p\) between the vertex and the focus can be calculated as follows:
- The vertex is at \((0, 2)\) and the focus is at \((3, 4)\).
- The distance in the x-direction is \(3 - 0 = 3\).
- The distance in the y-direction is \(4 - 2 = 2\).

Using the distance formula:
\[ p = \sqrt{(3 - 0)^2 + (4 - 2)^2} = \sqrt{9 + 4} = \sqrt{13} \]
However, since we are dealing with a parabola that opens towards the right, we will consider the horizontal distance \(p\) as \(3\).

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

Substituting \(h = 0\), \(k = 2\), and \(p = 3\):
\[ (y - 2)^2 = 4(3)(x - 0) \]
This simplifies to:
\[ (y - 2)^2 = 12x \]

### Step 5: Rearranging the equation.
To express this in a more standard form:
\[ y^2 - 4y + 4 = 12x \]
Rearranging gives:
\[ y^2 - 4y - 12x + 4 = 0 \]

### Final Equation:
Thus, the equation of the parabola is:
\[ y^2 - 4y - 12x + 4 = 0 \]