The equation `x^(2) - 2xy +y^(2) +3x +2 = 0` represents

Options

A. A parabola
B. An ellipse
C. A hyperbola
D. A circle

Explanation

To determine the type of conic section represented by the equation \(x^2 - 2xy + y^2 + 3x + 2 = 0\), we can follow these steps:

### Step 1: Identify the coefficients
The general form of a conic section is given by:
\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
From the given equation, we can identify the coefficients:
- \(A = 1\) (coefficient of \(x^2\))
- \(B = -2\) (coefficient of \(xy\))
- \(C = 1\) (coefficient of \(y^2\))
- \(D = 3\) (coefficient of \(x\))
- \(E = 0\) (coefficient of \(y\))
- \(F = 2\) (constant term)

### Step 2: Calculate the discriminant
The discriminant \(\Delta\) for conic sections is calculated using the formula:
\[ \Delta = B^2 - 4AC \]
Substituting the values we found:
\[ \Delta = (-2)^2 - 4(1)(1) = 4 - 4 = 0 \]

### Step 3: Determine the type of conic section
Next, we check the value of \(\Delta\):
- If \(\Delta < 0\), the conic is an ellipse.
- If \(\Delta = 0\), the conic is a parabola.
- If \(\Delta > 0\), the conic is a hyperbola.

Since we found \(\Delta = 0\), we conclude that the equation represents a **parabola**.

### Step 4: Verify using \(h^2 - ab\)
We can also verify by calculating \(h^2 - ab\):
- Here, \(h = \frac{B}{2} = \frac{-2}{2} = -1\)
- Thus, \(h^2 = (-1)^2 = 1\)
- \(ab = A \cdot C = 1 \cdot 1 = 1\)

Now, we check:
\[ h^2 - ab = 1 - 1 = 0 \]
Since \(h^2 - ab = 0\), this confirms that the conic section is indeed a **parabola**.

### Conclusion
The equation \(x^2 - 2xy + y^2 + 3x + 2 = 0\) represents a **parabola**.

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