The value of `lambda` for which the equation `x^2-y^2 - x - lambda y - 2 = 0` represent a pair of straight line, are

Options

A. `-3,1`
B. `3,-3`
C. `-1,1`
D. `3,1`

Answer

To find the value of \( \lambda \) for which the equation \( x^2 - y^2 - x - \lambda y - 2 = 0 \) represents a pair of straight lines, we can follow these steps:

### Step 1: Identify the coefficients
The given equation can be compared with the general form of a second-degree equation:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]
From the equation \( x^2 - y^2 - x - \lambda y - 2 = 0 \), we can identify the coefficients:
- \( a = 1 \)
- \( b = -1 \)
- \( c = -2 \)
- \( g = -\frac{1}{2} \)
- \( f = -\frac{\lambda}{2} \)
- \( h = 0 \)

### Step 2: Use the condition for a pair of straight lines
For the equation to represent a pair of straight lines, the following condition must hold:
\[ abc + 2fgh - af^2 - bg^2 - ch^2 = 0 \]

### Step 3: Substitute the coefficients into the condition
Substituting the identified coefficients into the condition:
\[ 1 \cdot (-1) \cdot (-2) + 2 \cdot \left(-\frac{\lambda}{2}\right) \cdot 0 \cdot 0 - 1 \cdot \left(-\frac{\lambda}{2}\right)^2 - (-1) \cdot \left(-\frac{1}{2}\right)^2 - (-2) \cdot 0^2 = 0 \]
This simplifies to:
\[ 2 - \frac{\lambda^2}{4} - \frac{1}{4} = 0 \]

### Step 4: Simplify the equation
Now, we simplify the equation:
\[ 2 - \frac{1}{4} - \frac{\lambda^2}{4} = 0 \]
\[ \frac{8}{4} - \frac{1}{4} - \frac{\lambda^2}{4} = 0 \]
\[ \frac{7}{4} - \frac{\lambda^2}{4} = 0 \]

### Step 5: Solve for \( \lambda^2 \)
Multiplying through by 4 to eliminate the denominator:
\[ 7 - \lambda^2 = 0 \]
\[ \lambda^2 = 7 \]

### Step 6: Find \( \lambda \)
Taking the square root of both sides gives:
\[ \lambda = \pm \sqrt{7} \]

### Final Answer
Thus, the values of \( \lambda \) for which the equation represents a pair of straight lines are:
\[ \lambda = \sqrt{7} \quad \text{and} \quad \lambda = -\sqrt{7} \]

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