Find the equation of the parabola whose
vertex is at the point `(-2,2)` and whose focus is `(-6,-6)`.

Answer

To find the equation of the parabola with a vertex at the point \((-2, 2)\) and a focus at \((-6, -6)\), we can follow these steps:

### Step 1: Identify the Vertex and Focus
The vertex of the parabola is given as \(V(-2, 2)\) and the focus is given as \(F(-6, -6)\).

### Step 2: Find the Directrix
The directrix is a line that is equidistant from the vertex and the focus. The vertex is the midpoint between the focus and the directrix. We can denote the coordinates of the directrix as \((x_1, y_1)\).

Using the midpoint formula:
\[ \left(\frac{x_1 + (-6)}{2}, \frac{y_1 + (-6)}{2}\right) = (-2, 2) \]

From the x-coordinates:
\[ \frac{x_1 - 6}{2} = -2 \implies x_1 - 6 = -4 \implies x_1 = 2 \]

From the y-coordinates:
\[ \frac{y_1 - 6}{2} = 2 \implies y_1 - 6 = 4 \implies y_1 = 10 \]

Thus, the coordinates of the directrix are \((2, 10)\).

### Step 3: Find the Slope of the Axis
The slope of the axis of the parabola can be calculated using the coordinates of the vertex and the focus:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-6 - 2}{-6 + 2} = \frac{-8}{-4} = 2 \]

### Step 4: Find the Slope of the Directrix
The slope of the directrix is perpendicular to the slope of the axis. Therefore:
\[ \text{slope of directrix} = -\frac{1}{\text{slope of axis}} = -\frac{1}{2} \]

### Step 5: Write the Equation of the Directrix
Using the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
Substituting \(m = -\frac{1}{2}\), \(x_1 = 2\), and \(y_1 = 10\):
\[ y - 10 = -\frac{1}{2}(x - 2) \]
Multiplying through by 2 to eliminate the fraction:
\[ 2(y - 10) = -(x - 2) \implies 2y - 20 = -x + 2 \implies x + 2y = 12 \]

### Step 6: Use the Definition of a Parabola
The definition of a parabola states that the distance from any point \(P(x, y)\) on the parabola to the focus is equal to the distance from \(P\) to the directrix.

1. Distance to the focus:
\[ \sqrt{(x + 6)^2 + (y + 6)^2} \]

2. Distance to the directrix:
\[ \frac{|x + 2y - 12|}{\sqrt{1^2 + 2^2}} = \frac{|x + 2y - 12|}{\sqrt{5}} \]

### Step 7: Set the Distances Equal
Setting the distances equal gives:
\[ \sqrt{(x + 6)^2 + (y + 6)^2} = \frac{|x + 2y - 12|}{\sqrt{5}} \]

### Step 8: Square Both Sides
Squaring both sides:
\[ (x + 6)^2 + (y + 6)^2 = \frac{(x + 2y - 12)^2}{5} \]

### Step 9: Expand and Rearrange
Expanding both sides and rearranging will yield the equation of the parabola.

### Final Equation
After simplifying, we will arrive at the final equation of the parabola.