The focus of the parabola `x^(2)-2x-8y-23=0` is

Options

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

Answer

To find the focus of the parabola given by the equation \( x^2 - 2x - 8y - 23 = 0 \), we can follow these steps:

### Step 1: Rearrange the equation
Start by rearranging the equation to isolate the \( y \) terms. We can do this by moving all other terms to the right side.

\[ x^2 - 2x = 8y + 23 \]

### Step 2: Complete the square for the \( x \) terms
To complete the square for the \( x \) terms, we take the coefficient of \( x \), which is -2, divide it by 2 to get -1, and then square it to get 1. We add and subtract this value on the left side.

\[ x^2 - 2x + 1 - 1 = 8y + 23 \]

This simplifies to:

\[ (x - 1)^2 - 1 = 8y + 23 \]

### Step 3: Simplify the equation
Now, we can simplify the equation further:

\[ (x - 1)^2 = 8y + 24 \]

### Step 4: Rearrange to standard form
Next, we rearrange the equation to the standard form of a parabola:

\[ (x - 1)^2 = 8(y + 3) \]

### Step 5: Identify parameters
From the standard form \( (x - h)^2 = 4a(y - k) \), we can identify:

- \( h = 1 \)
- \( k = -3 \)
- \( 4a = 8 \) which gives \( a = 2 \)

### Step 6: Find the focus
The focus of a parabola in this form is given by the coordinates \( (h, k + a) \):

\[ \text{Focus} = (1, -3 + 2) = (1, -1) \]

### Final Answer
Thus, the focus of the parabola is \( (1, -1) \).

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