Let `x^(2)+3xy+2y^(2)+2x+3y=0` a hyperbola, then which of the follwing is true ?

Options

A. equation of asymptotes is `x^(2)+3xy+2y^(2)+2x+3y+1=0`
B. Equations of asymptotes are `x+y+1=0 and x+2y+1=0`
C. The equation of conjugate hyperbola is `x^(2)+3xy+2y^(2)+2x+3y+2=0`
D. none

Answer

To determine the properties of the hyperbola given by the equation \( x^2 + 3xy + 2y^2 + 2x + 3y = 0 \), we will 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:
- \( A = 1 \)
- \( B = 3 \)
- \( C = 2 \)
- \( D = 2 \)
- \( E = 3 \)
- \( F = 0 \)

### Step 2: Determine the type of conic
To determine the type of conic, we calculate the discriminant \( \Delta \):
\[ \Delta = B^2 - 4AC \]
Substituting the values:
\[ \Delta = 3^2 - 4 \cdot 1 \cdot 2 = 9 - 8 = 1 \]
Since \( \Delta > 0 \), the conic is a hyperbola.

### Step 3: Find the asymptotes
The asymptotes of a hyperbola can be found using the formula:
\[ y = mx + c \]
Where \( m \) is determined from the coefficients of the conic. The equations of the asymptotes can be derived from the quadratic form:
\[ Ax^2 + Bxy + Cy^2 = 0 \]
This leads to the characteristic equation:
\[ C \cdot m^2 + B \cdot m + A = 0 \]
Substituting our values:
\[ 2m^2 + 3m + 1 = 0 \]
Using the quadratic formula \( m = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \):
\[ m = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2} = \frac{-3 \pm \sqrt{9 - 8}}{4} = \frac{-3 \pm 1}{4} \]
This gives us:
\[ m_1 = \frac{-2}{4} = -\frac{1}{2}, \quad m_2 = \frac{-4}{4} = -1 \]
Thus, the equations of the asymptotes are:
\[ y = -\frac{1}{2}x + c_1 \quad \text{and} \quad y = -x + c_2 \]

### Step 4: Determine the conjugate parabola
The conjugate parabola can be derived from the hyperbola's equation. The relationship is given by:
\[ 2a - h = 0 \]
Where \( h \) is derived from the hyperbola's equation. To find \( h \), we can rearrange the hyperbola's equation:
\[ h = \frac{D^2}{4A} + \frac{E^2}{4C} - F \]
Substituting in the values:
\[ h = \frac{2^2}{4 \cdot 1} + \frac{3^2}{4 \cdot 2} - 0 = 1 + \frac{9}{8} = \frac{8}{8} + \frac{9}{8} = \frac{17}{8} \]
Thus, we can find the conjugate parabola equation:
\[ 2a - \frac{17}{8} = 0 \implies 2a = \frac{17}{8} \implies a = \frac{17}{16} \]

### Conclusion
From the calculations, we can conclude that the hyperbola has asymptotes and a conjugate parabola derived from the original equation. The properties of the hyperbola and its asymptotes can be summarized as follows:
- The hyperbola is confirmed by the discriminant.
- The equations of the asymptotes are derived.
- The conjugate parabola is also determined.