The line x+y+1=0 is an asymptote of `x^(2)-y^(2)+x-y-2=0`. The other asymptote is

Options

A. a) x+y=0
B. b) x-y=0
C. c) x-y=1
D. d) x-y+1=0

Answer

To find the other asymptote of the hyperbola defined by the equation \(x^2 - y^2 + x - y - 2 = 0\), given that one asymptote is \(x + y + 1 = 0\), we can follow these steps:

### Step 1: Rearrange the hyperbola equation
We start with the equation of the hyperbola:
\[ x^2 - y^2 + x - y - 2 = 0 \]
Rearranging gives:
\[ x^2 + x - y^2 - y - 2 = 0 \]

### Step 2: Complete the square
We will complete the square for both \(x\) and \(y\).

For \(x^2 + x\):
\[ x^2 + x = (x + \frac{1}{2})^2 - \frac{1}{4} \]

For \(-y^2 - y\):
\[ -y^2 - y = -\left(y^2 + y\right) = -\left((y + \frac{1}{2})^2 - \frac{1}{4}\right) = - (y + \frac{1}{2})^2 + \frac{1}{4} \]

Substituting these back into the equation:
\[ \left(x + \frac{1}{2}\right)^2 - \frac{1}{4} - \left(y + \frac{1}{2}\right)^2 + \frac{1}{4} - 2 = 0 \]
This simplifies to:
\[ \left(x + \frac{1}{2}\right)^2 - \left(y + \frac{1}{2}\right)^2 - 2 = 0 \]

### Step 3: Rearranging to standard form
Rearranging gives:
\[ \left(x + \frac{1}{2}\right)^2 - \left(y + \frac{1}{2}\right)^2 = 2 \]
Dividing through by 2:
\[ \frac{\left(x + \frac{1}{2}\right)^2}{2} - \frac{\left(y + \frac{1}{2}\right)^2}{2} = 1 \]

### Step 4: Identify the center and slopes of the asymptotes
The center of the hyperbola is at \((-1/2, -1/2)\). The slopes of the asymptotes for a hyperbola of the form \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\) are given by:
\[ y - k = \pm \frac{b}{a}(x - h) \]
In our case, since \(a^2 = 2\) and \(b^2 = 2\), the slopes are:
\[ \pm 1 \]

### Step 5: Write the equations of the asymptotes
The first asymptote is given as \(x + y + 1 = 0\) (which has a slope of -1). The second asymptote, with a slope of 1, can be written as:
\[ y + \frac{1}{2} = 1\left(x + \frac{1}{2}\right) \]
This simplifies to:
\[ y = x \]

### Step 6: Final form of the other asymptote
Thus, the equation of the other asymptote is:
\[ x - y = 0 \]

### Conclusion
The other asymptote is \(x - y = 0\).