Find the asymptotes of the hyperola. `2x^(2)-xy-y^(2)+2x-2y+2=0`

Answer

To find the asymptotes of the hyperbola given by the equation \(2x^2 - xy - y^2 + 2x - 2y + 2 = 0\), we will follow these steps:

### Step 1: Rewrite the Equation
We start with the equation of the hyperbola:
\[ 2x^2 - xy - y^2 + 2x - 2y + 2 = 0 \]
To find the asymptotes, we modify this equation by introducing a parameter \(\lambda\):
\[ 2x^2 - xy - y^2 + 2x - 2y + \lambda = 0 \]

### Step 2: Identify Coefficients
We can identify the coefficients \(a\), \(b\), \(c\), \(g\), \(f\), and \(h\) from the general form of the conic section:
\[ ax^2 + bxy + cy^2 + 2gx + 2fy + c = 0 \]
From our modified equation, we have:
- \(a = 2\)
- \(b = -1\)
- \(c = -1\)
- \(g = 1\) (since \(2g = 2\))
- \(f = -1\) (since \(2f = -2\))
- \(c = \lambda\)

### Step 3: Use the Condition for Pair of Straight Lines
The condition for the equation to represent a pair of straight lines is given by:
\[ abc + 2fgh - af^2 - bg^2 - ch^2 = 0 \]
Substituting the values we found:
\[ (2)(-1)(\lambda) + 2(-1)(1)(-\frac{1}{2}) - (2)(-1)^2 - (-1)(1)^2 - \lambda(-\frac{1}{2})^2 = 0 \]

### Step 4: Simplify the Equation
Now, we simplify the equation:
\[ -2\lambda + 2 \cdot \frac{1}{2} - 2 - 1 - \frac{\lambda}{4} = 0 \]
This simplifies to:
\[ -2\lambda + 1 - 2 - 1 - \frac{\lambda}{4} = 0 \]
Combining like terms:
\[ -2\lambda - \frac{\lambda}{4} = 0 \]
Finding a common denominator:
\[ -\frac{8\lambda}{4} - \frac{\lambda}{4} = 0 \]
This gives:
\[ -\frac{9\lambda}{4} = 0 \]
Thus, we find:
\[ \lambda = 0 \]

### Step 5: Write the Equation of Asymptotes
Now substituting \(\lambda = 0\) back into the modified equation gives us the equation of the asymptotes:
\[ 2x^2 - xy - y^2 + 2x - 2y = 0 \]

### Step 6: Factor the Asymptotes
To find the lines, we can factor this equation:
\[ 2x^2 - xy - y^2 + 2x - 2y = 0 \]
We can rearrange and factor it:
\[ (2x + y)(x - 2y) = 0 \]
Thus, the asymptotes are:
\[ y = -2x \quad \text{and} \quad y = \frac{1}{2}x \]

### Final Answer
The asymptotes of the hyperbola are:
1. \(y = -2x\)
2. \(y = \frac{1}{2}x\)