The coordinates of the vertex of the parabola `y^(2)=4(x+y)` is

Options

A. (0,0)
B. (2,1)
C. (-1,2)
D. (1,2)

Answer

To find the coordinates of the vertex of the parabola given by the equation \( y^2 = 4(x + y) \), we will follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ y^2 = 4(x + y) \]
Rearranging it gives:
\[ y^2 - 4y = 4x \]

### Step 2: Complete the square for \( y \)
To complete the square for the left-hand side, we take the coefficient of \( y \), which is \(-4\), halve it to get \(-2\), and then square it to get \(4\). We add and subtract \(4\) on the left side:
\[ y^2 - 4y + 4 - 4 = 4x \]
This simplifies to:
\[ (y - 2)^2 - 4 = 4x \]
Now, add \(4\) to both sides:
\[ (y - 2)^2 = 4x + 4 \]

### Step 3: Factor the right-hand side
Factor the right-hand side:
\[ (y - 2)^2 = 4(x + 1) \]

### Step 4: Identify the vertex
The equation is now in the standard form of a parabola:
\[ (y - k)^2 = 4a(x - h) \]
where \( (h, k) \) is the vertex. From our equation, we can identify:
- \( k = 2 \)
- \( h = -1 \)

Thus, the coordinates of the vertex are:
\[ (h, k) = (-1, 2) \]

### Final Answer
The coordinates of the vertex of the parabola are \((-1, 2)\).

---