The second degree equation `x^(2) + 4y - 2x -4y + 2= 0` represents

Options

A. A point
B. An ellipse of semi-major axis 1
C. An ellipse with eccentricity `(sqrt3)/(2)`
D. None of the above

Answer

To determine what the second degree equation \( x^2 + 4y - 2x - 4y + 2 = 0 \) represents, we can follow these steps:

### Step 1: Simplify the Equation
Start with the given equation:
\[ x^2 + 4y - 2x - 4y + 2 = 0 \]
Combine like terms:
\[ x^2 - 2x + 2 = 0 \]

### Step 2: Rearrange the Equation
Rearranging the equation gives:
\[ x^2 - 2x + 2 = 0 \]

### Step 3: Complete the Square
To complete the square for the \( x \) terms, we take the coefficient of \( x \), which is -2, halve it to get -1, and then square it to get 1. We can rewrite the equation:
\[ (x - 1)^2 + 1 = 0 \]

### Step 4: Analyze the Equation
Now we can analyze the equation:
\[ (x - 1)^2 + 1 = 0 \]
This implies:
\[ (x - 1)^2 = -1 \]
Since the square of a real number cannot be negative, this indicates that there are no real solutions.

### Step 5: Identify the Representation
The equation \( (x - 1)^2 + (y - \frac{1}{2})^2 = 0 \) represents a single point in the coordinate system. The point is \( (1, \frac{1}{2}) \).

### Conclusion
Thus, the second degree equation represents a point, specifically the point \( P(1, \frac{1}{2}) \).

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