The focus of the parabola `y^(2) - x - 2y + 2 = 0 ` is (i) `((5)/( 4), 1)` (ii) `((1)/(4),1)` (iii) `((3)/(4),1)` (iv) (1,1)

Options

A. `((5)/( 4), 1)`
B. `((1)/(4),1)`
C. `((3)/(4),1)`
D. (1,1)

Answer

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

### Step 1: Rearranging the Equation
First, we need to rearrange the equation into a standard form. Start with the original equation:
\[ y^2 - x - 2y + 2 = 0 \]
Rearranging gives:
\[ y^2 - 2y + 2 = x \]

### Step 2: Completing the Square
Next, we will complete the square for the \( y \) terms:
\[ y^2 - 2y + 1 = x - 1 \]
This simplifies to:
\[ (y - 1)^2 = x - 1 \]

### Step 3: Identifying the Standard Form
Now, we can rewrite the equation in the standard form of a parabola:
\[ (y - 1)^2 = 1(x - 1) \]
This indicates that the parabola opens to the right and is centered at the point \( (1, 1) \).

### Step 4: Finding the Focus
For a parabola in the form \( (y - k)^2 = 4p(x - h) \), the focus is given by the point \( (h + p, k) \). Here, \( h = 1 \), \( k = 1 \), and \( 4p = 1 \) which implies \( p = \frac{1}{4} \).

Thus, the focus is:
\[ (h + p, k) = \left(1 + \frac{1}{4}, 1\right) = \left(\frac{5}{4}, 1\right) \]

### Conclusion
The focus of the parabola \( y^2 - x - 2y + 2 = 0 \) is:
\[ \left(\frac{5}{4}, 1\right) \]

### Answer
The correct option is (i) \( \left(\frac{5}{4}, 1\right) \).

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