`y^2-2x-2y+5=0` represents

Options

A. A pair of straight line
B. A circle with centre (1, 1)
C. A parabola with vetex (2, 1)
D. A parabola with directrix ` x= (3)/(2)`

Answer

To determine what the equation \( y^2 - 2x - 2y + 5 = 0 \) represents, we will manipulate the equation step by step.

### Step 1: Rearranging the Equation
Start by rearranging the equation to isolate the terms involving \( y \) on one side and the terms involving \( x \) on the other side.

\[ y^2 - 2y = 2x - 5 \]

### Step 2: Completing the Square
Next, we will complete the square for the left side of the equation. To do this, we take the coefficient of \( y \), which is -2, halve it to get -1, and then square it to get 1. We will add and subtract this value.

\[ y^2 - 2y + 1 - 1 = 2x - 5 \]

This simplifies to:

\[ (y - 1)^2 - 1 = 2x - 5 \]

Now, we can add 1 to both sides:

\[ (y - 1)^2 = 2x - 4 \]

### Step 3: Factoring the Right Side
Next, we can factor the right side:

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

### Step 4: Identifying the Conic Section
Now, we can compare this equation with the standard form of a parabola, which is:

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

From our equation, we can identify:
- \( k = 1 \)
- \( h = 2 \)
- \( 4a = 2 \) which implies \( a = \frac{1}{2} \)

### Conclusion
Thus, the equation \( y^2 - 2x - 2y + 5 = 0 \) represents a parabola. The vertex of the parabola is at the point \( (h, k) = (2, 1) \).