The focus of the parabola `y^(2)-4y -x + 3 =0` is

Options

A. `((3)/(4), 2)`
B. `((3)/(4), -2)`
C. `(2, (-3)/(4))`
D. `((-3)/(4), 2)`

Answer

To find the focus of the parabola given by the equation \( y^2 - 4y - x + 3 = 0 \), we will follow these steps:

### Step 1: Rearrange the equation
Start by rearranging the equation to isolate \( x \):
\[ y^2 - 4y + 3 = x \]

### Step 2: Complete the square for \( y \)
Next, we need to complete the square for the \( y \) terms in the equation \( y^2 - 4y \):
\[ y^2 - 4y = (y - 2)^2 - 4 \]
So, substituting this back into the equation gives:
\[ (y - 2)^2 - 4 + 3 = x \]
This simplifies to:
\[ (y - 2)^2 - 1 = x \]
or
\[ (y - 2)^2 = x + 1 \]

### Step 3: Identify the standard form of the parabola
Now, we can rewrite the equation in the standard form of a parabola:
\[ (y - 2)^2 = 1(x + 1) \]
This is in the form \( (y - k)^2 = 4p(x - h) \), where \( (h, k) \) is the vertex and \( p \) is the distance from the vertex to the focus.

### Step 4: Determine the vertex and focus
From the equation \( (y - 2)^2 = 1(x + 1) \):
- The vertex \( (h, k) \) is \( (-1, 2) \).
- Here, \( 4p = 1 \), so \( p = \frac{1}{4} \).

Since the parabola opens to the right, the focus will be at:
\[ (h + p, k) = \left(-1 + \frac{1}{4}, 2\right) = \left(-\frac{3}{4}, 2\right) \]

### Final Answer
The focus of the parabola is:
\[ \left(-\frac{3}{4}, 2\right) \]

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