The equation `x^(2) - 2xy +y^(2) +3x +2 = 0` represents (a) a parabola (b) an ellipse (c) a hyperbola (d) a circle

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 will compare it with the general form of a conic section equation.

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

### Step 2: Calculate the discriminant
The discriminant \( \Delta \) of the conic section is given by:
\[ \Delta = B^2 - 4AC \]
Substituting the values of \( A \), \( B \), and \( C \):
\[ \Delta = (-2)^2 - 4(1)(1) = 4 - 4 = 0 \]

### Step 3: Determine the type of conic section
The type of conic section can be determined based on the value of the discriminant \( \Delta \):
- 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 \( \Delta = 0 \), this indicates that the equation represents a parabola.

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

The correct answer is:
**(a) a parabola**

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