The focus of the parabola `x^2-8x+2y+7=0` is

Options

A. `(0,-1/2)`
B. (4,4)
C. `(4,9/2)`
D. `(-4,-9/2)`

Answer

To find the focus of the parabola given by the equation \( x^2 - 8x + 2y + 7 = 0 \), we will follow these steps:

### Step 1: Rearrange the equation
Start by rearranging the equation to isolate \( y \):
\[ x^2 - 8x + 2y + 7 = 0 \]
This can be rewritten as:
\[ 2y = -x^2 + 8x - 7 \]
Now, divide everything by 2:
\[ y = -\frac{1}{2}x^2 + 4x - \frac{7}{2} \]

### Step 2: Complete the square for the \( x \) terms
We will complete the square for the quadratic expression in \( x \):
\[ y = -\frac{1}{2}(x^2 - 8x) - \frac{7}{2} \]
To complete the square, take half of the coefficient of \( x \) (which is -8), square it, and add/subtract it inside the parentheses:
\[ y = -\frac{1}{2}((x - 4)^2 - 16) - \frac{7}{2} \]
This simplifies to:
\[ y = -\frac{1}{2}(x - 4)^2 + 8 - \frac{7}{2} \]
Calculating \( 8 - \frac{7}{2} \):
\[ 8 = \frac{16}{2} \implies 8 - \frac{7}{2} = \frac{16}{2} - \frac{7}{2} = \frac{9}{2} \]
Thus, we have:
\[ y = -\frac{1}{2}(x - 4)^2 + \frac{9}{2} \]

### Step 3: Identify the vertex and orientation
The equation is now in the form:
\[ y = -\frac{1}{2}(x - 4)^2 + \frac{9}{2} \]
From this, we can identify:
- The vertex \( (h, k) \) is \( (4, \frac{9}{2}) \).
- The parabola opens downwards since the coefficient of \( (x - 4)^2 \) is negative.

### Step 4: Find the focus
For a parabola in the form \( y = a(x - h)^2 + k \), the focus can be found using:
- The distance \( a \) from the vertex to the focus is given by \( a = \frac{1}{4p} \), where \( p \) is the distance from the vertex to the focus. Here, \( a = -\frac{1}{2} \), so:
\[ 4p = -2 \implies p = -\frac{1}{2} \]
The focus is located at \( (h, k + p) \):
\[ \text{Focus} = \left(4, \frac{9}{2} - \frac{1}{2}\right) = \left(4, \frac{8}{2}\right) = (4, 4) \]

### Final Answer
The focus of the parabola is \( (4, 4) \).

---