If `ax^2 + 4xy + y^2 + ax + 3y +2=0` represents a parabola, then a =

Options

A. `-4`
B. `4`
C. `0`
D. none

Answer

To determine the value of \( a \) such that the equation \( ax^2 + 4xy + y^2 + ax + 3y + 2 = 0 \) represents a parabola, we will analyze the coefficients of the general conic section equation and apply the conditions for it to be a parabola.

### Step-by-Step Solution:

1. **Identify the Coefficients:**
The given equation can be compared to the general conic section equation:
\[ Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0 \]
From the equation \( ax^2 + 4xy + y^2 + ax + 3y + 2 = 0 \), we identify:
- \( A = a \)
- \( H = 2 \) (since \( 2H = 4 \))
- \( B = 1 \)
- \( G = \frac{a}{2} \)
- \( F = \frac{3}{2} \)
- \( C = 2 \)

2. **Apply the Conditions for a Parabola:**
For the conic section to represent a parabola, the following conditions must hold:
- The discriminant \( \Delta \) must equal zero:
\[ \Delta = ABC + 2FGH - AF^2 - BG^2 - H^2 = 0 \]
- The condition \( H^2 = AB \) must also be satisfied.

3. **Calculate the Discriminant:**
Substitute the values of \( A, B, C, G, F, H \):
\[ \Delta = a \cdot 1 \cdot 2 + 2 \cdot \frac{3}{2} \cdot 2 - a \left(\frac{3}{2}\right)^2 - 1 \left(\frac{a}{2}\right)^2 - 2^2 \]
Simplifying this gives:
\[ \Delta = 2a + 6 - \frac{9a}{4} - \frac{a^2}{4} - 4 \]
Combine like terms:
\[ \Delta = 2a - 4 + 6 - \frac{9a}{4} - \frac{a^2}{4} \]
\[ \Delta = 2a + 2 - 4 - \frac{9a}{4} - \frac{a^2}{4} \]
\[ \Delta = -\frac{a^2}{4} + \left(2 - \frac{9}{4}\right)a + 2 \]
\[ \Delta = -\frac{a^2}{4} - \frac{1}{4}a + 2 \]

4. **Set the Discriminant to Zero:**
Set \( \Delta = 0 \):
\[ -\frac{a^2}{4} - \frac{1}{4}a + 2 = 0 \]
Multiply through by -4 to eliminate the fraction:
\[ a^2 + a - 8 = 0 \]

5. **Solve the Quadratic Equation:**
Use the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ a = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-8)}}{2 \cdot 1} \]
\[ a = \frac{-1 \pm \sqrt{1 + 32}}{2} \]
\[ a = \frac{-1 \pm \sqrt{33}}{2} \]

6. **Conclusion:**
The values of \( a \) that make the equation represent a parabola are:
\[ a = \frac{-1 + \sqrt{33}}{2} \quad \text{or} \quad a = \frac{-1 - \sqrt{33}}{2} \]