Co-ordinates of the focus of the parabola `x^2 - 4x-8y – 4=0` are

Options

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

Answer

To find the coordinates of the focus of the parabola given by the equation \(x^2 - 4x - 8y - 4 = 0\), we will follow these steps:

### Step 1: Rearrange the equation
Start by rearranging the given equation to isolate the \(y\) term.

\[ x^2 - 4x - 4 = 8y \]

### Step 2: Complete the square for the \(x\) terms
Next, we will complete the square for the \(x\) terms on the left side of the equation.

1. Take the coefficient of \(x\) which is \(-4\), halve it to get \(-2\), and square it to get \(4\).
2. Add and subtract \(4\) inside the equation:

\[ (x^2 - 4x + 4 - 4) - 4 = 8y \]

This simplifies to:

\[ (x - 2)^2 - 4 = 8y \]

### Step 3: Rearrange to standard form
Now, rearranging gives us:

\[ (x - 2)^2 = 8y + 4 \]

This can be rewritten as:

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

### Step 4: Identify the parameters of the parabola
Now, we can identify the standard form of the parabola, which is given by:

\[ (x - h)^2 = 4a(y - k) \]

From our equation, we see:

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

### Step 5: Find the coordinates of the focus
The coordinates of the focus of a parabola in this form are given by \((h, k + a)\).

Substituting the values we found:

- \(h = 2\)
- \(k = -1\)
- \(a = 2\)

Thus, the coordinates of the focus are:

\[ (2, -1 + 2) = (2, 1) \]

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

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