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

Options

A. a parabola
B. a pair of straight line
C. an ellipse
D. a hyperbola

Explanation

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

### Step 1: Rewrite the equation in standard form
We start with the given equation:
\[ x^2 + 4y^2 - 2x - 4y + 2 = 0 \]
We can rearrange it to group the \( x \) and \( y \) terms:
\[ x^2 - 2x + 4y^2 - 4y + 2 = 0 \]

### Step 2: Identify coefficients
We can identify the coefficients from the general form of a second degree equation:
\[ ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0 \]
From our equation:
- \( a = 1 \)
- \( b = 4 \)
- \( h = 0 \) (since there is no \( xy \) term)
- \( g = -1 \) (from \( -2x \))
- \( f = -2 \) (from \( -4y \))
- \( c = 2 \)

### Step 3: Calculate the discriminant
The discriminant \( \Delta \) is given by the formula:
\[ \Delta = abc + 2hfg - af^2 - bg^2 - h^2 \]
Substituting the values we found:
\[ \Delta = (1)(4)(2) + 2(0)(-2)(-1) - (1)(-2)^2 - (4)(-1)^2 - (0)^2 \]
Calculating each term:
- \( abc = 8 \)
- \( 2hfg = 0 \)
- \( af^2 = 4 \)
- \( bg^2 = 4 \)
- \( h^2 = 0 \)

Putting it all together:
\[ \Delta = 8 + 0 - 4 - 4 - 0 = 0 \]

### Step 4: Calculate \( h^2 - ab \)
Next, we calculate \( h^2 - ab \):
\[ h^2 - ab = 0^2 - (1)(4) = 0 - 4 = -4 \]

### Step 5: Analyze the results
We have:
- \( \Delta = 0 \)
- \( h^2 - ab < 0 \)

According to the properties of conic sections:
- If \( \Delta = 0 \) and \( h^2 - ab < 0 \), the equation represents an ellipse.

### Conclusion
Thus, the equation \( x^2 + 4y^2 - 2x - 4y + 2 = 0 \) represents an **ellipse**.

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