The lines represented by the equation `x^(2)+2sqrt(3)xy+3y^(2)-3x-3sqrt(3)y-4=0` are

Options

A. Perpendicular to each other
B. Parallel
C. Inclined at `45^(@)` to each other
D. Inclined at `60^(@)` to each other

Explanation

To solve the problem, we need to determine the angle between the two lines represented by the equation:

\[ x^2 + 2\sqrt{3}xy + 3y^2 - 3x - 3\sqrt{3}y - 4 = 0 \]

### Step 1: Identify coefficients
We can compare the given equation with the general form of the conic section:

\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]

From the given equation, we identify:
- \( A = 1 \)
- \( B = 2\sqrt{3} \)
- \( C = 3 \)
- \( D = -3 \)
- \( E = -3\sqrt{3} \)
- \( F = -4 \)

### Step 2: Calculate \( h \), \( g \), \( f \), and \( c \)
Using the relationships:
- \( 2h = B \)
- \( 2g = D \)
- \( 2f = E \)
- \( c = F \)

We find:
- \( h = \frac{B}{2} = \frac{2\sqrt{3}}{2} = \sqrt{3} \)
- \( g = \frac{D}{2} = \frac{-3}{2} = -\frac{3}{2} \)
- \( f = \frac{E}{2} = \frac{-3\sqrt{3}}{2} \)
- \( c = -4 \)

### Step 3: Use the formula for the angle between the lines
The angle \( \theta \) between the two lines can be calculated using the formula:

\[ \tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b} \]

Where:
- \( a = A = 1 \)
- \( b = C = 3 \)

### Step 4: Substitute values into the formula
First, calculate \( h^2 - ab \):
- \( h^2 = (\sqrt{3})^2 = 3 \)
- \( ab = 1 \cdot 3 = 3 \)

Thus:
\[ h^2 - ab = 3 - 3 = 0 \]

Now substitute into the formula:
\[ \tan \theta = \frac{2\sqrt{0}}{1 + 3} = \frac{0}{4} = 0 \]

### Step 5: Determine the angle
Since \( \tan \theta = 0 \), we find:
\[ \theta = \tan^{-1}(0) = 0 \]

### Conclusion
The angle between the two lines is \( 0 \) degrees, indicating that the lines are parallel.