The mirror image of the focus to the parabola `4(x + y) = y^(2)` w.r.t. the directrix is

Options

A. (0, 2)
B. (2, 2)
C. (-4, 2)
D. (-2, 2)

Answer

To find the mirror image of the focus of the parabola \(4(x + y) = y^2\) with respect to the directrix, we will follow these steps:

### Step 1: Rewrite the Parabola in Standard Form
The given equation of the parabola is:
\[ 4(x + y) = y^2 \]
Rearranging it gives:
\[ y^2 - 4y = 4x \]
Completing the square for the \(y\) terms:
\[ y^2 - 4y + 4 = 4x + 4 \]
This simplifies to:
\[ (y - 2)^2 = 4(x + 1) \]
This is now in the standard form of a parabola, which is \((y - k)^2 = 4p(x - h)\), where \((h, k)\) is the vertex.

### Step 2: Identify the Vertex and Focus
From the equation \((y - 2)^2 = 4(x + 1)\), we can identify:
- Vertex \((h, k) = (-1, 2)\)
- The value of \(p = 1\) (since \(4p = 4\), thus \(p = 1\))

The focus of the parabola is located at:
\[ (h + p, k) = (-1 + 1, 2) = (0, 2) \]

### Step 3: Find the Directrix
The directrix of a parabola is given by the equation:
\[ x = h - p \]
Substituting the values we found:
\[ x = -1 - 1 = -2 \]

### Step 4: Find the Mirror Image of the Focus with Respect to the Directrix
The focus is at \((0, 2)\) and the directrix is the line \(x = -2\). To find the mirror image of the focus with respect to the directrix, we calculate the distance from the focus to the directrix:
\[ \text{Distance} = 0 - (-2) = 2 \]
The mirror image will be located at a distance of 2 units on the opposite side of the directrix:
\[ \text{Mirror Image} = -2 - 2 = -4 \]
Thus, the coordinates of the mirror image are:
\[ (-4, 2) \]

### Final Answer
The mirror image of the focus with respect to the directrix is:
\[ \boxed{(-4, 2)} \]