The equation `x^(2) + 4xy + 4y ^(2) - 3x - 6 = 0 ` represents

Options

A. A. a circle
B. B. a parabola
C. C. an ellipse
D. D. a hyperbola

Answer

To determine the type of conic section represented by the equation \( x^2 + 4xy + 4y^2 - 3x - 6 = 0 \), we will follow these steps:

### Step 1: Identify coefficients
The general form of a conic section is given by:
\[ ax^2 + bxy + cy^2 + 2gx + 2fy + c = 0 \]
From the given equation, we can identify the coefficients:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 4 \) (coefficient of \( xy \))
- \( c = 4 \) (coefficient of \( y^2 \))
- \( g = -\frac{3}{2} \) (since \( 2g = -3 \))
- \( f = 0 \) (since there is no \( y \) term)
- \( h = 2 \) (since \( 2h = 4 \))

### Step 2: Calculate the discriminant (\( \Delta \))
The discriminant for conic sections is given by:
\[ \Delta = abc + 2fgh - af^2 - bg^2 - ch^2 \]
Substituting the values we identified:
\[ \Delta = (1)(4)(-6) + 2(0)(-\frac{3}{2})(2) - (1)(0^2) - (4)(-\frac{3}{2})^2 - (4)(2^2) \]
Calculating each term:
- \( abc = 1 \cdot 4 \cdot -6 = -24 \)
- \( 2fgh = 2 \cdot 0 \cdot -\frac{3}{2} \cdot 2 = 0 \)
- \( -af^2 = -1 \cdot 0^2 = 0 \)
- \( -bg^2 = -4 \cdot \left(-\frac{3}{2}\right)^2 = -4 \cdot \frac{9}{4} = -9 \)
- \( -ch^2 = -4 \cdot 2^2 = -4 \cdot 4 = -16 \)

Now substituting these values into the discriminant:
\[ \Delta = -24 + 0 + 0 - 9 - 16 = -49 \]

### Step 3: Calculate \( h^2 - ab \)
Next, we calculate:
\[ h^2 - ab \]
Substituting the values:
\[ h^2 = 2^2 = 4 \]
\[ ab = 1 \cdot 4 = 4 \]
Thus,
\[ h^2 - ab = 4 - 4 = 0 \]

### Step 4: Determine the type of conic section
We know the following conditions:
- If \( \Delta \neq 0 \) and \( h^2 - ab = 0 \), then the conic section is a parabola.

From our calculations:
- \( \Delta = -49 \neq 0 \)
- \( h^2 - ab = 0 \)

Thus, the equation represents a **parabola**.

### Final Answer
The equation \( x^2 + 4xy + 4y^2 - 3x - 6 = 0 \) represents a parabola.

---