The equation `12x^(2)+7xy-12y^(2)-18x+y+6=0` represents

Options

A. a pair of straight lines
B. a parabola
C. an ellipse
D. a hyperbola

Explanation

To determine the type of conic section represented by the equation \(12x^2 + 7xy - 12y^2 - 18x + y + 6 = 0\), we will follow these steps:

### Step 1: Identify coefficients
We need to identify the coefficients \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\) from the general conic section equation:
\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
From the given equation:
- \(A = 12\)
- \(B = 7\)
- \(C = -12\)
- \(D = -18\)
- \(E = 1\)
- \(F = 6\)

### Step 2: Calculate the discriminant
The discriminant \(D\) for conic sections is given by the formula:
\[ D = B^2 - 4AC \]
Substituting the values:
\[ D = 7^2 - 4(12)(-12) = 49 + 576 = 625 \]

### Step 3: Determine the type of conic
Now we analyze the value of \(D\):
- If \(D < 0\), the conic is an ellipse (or a circle).
- If \(D = 0\), the conic is a parabola.
- If \(D > 0\), the conic is a hyperbola.

In our case:
\[ D = 625 > 0 \]
This indicates that the given equation represents a hyperbola.

### Conclusion
The equation \(12x^2 + 7xy - 12y^2 - 18x + y + 6 = 0\) represents a hyperbola.

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