If the equation `x^2-y^2-2x+2y+lamda=0` represent a degenerate conic . Find the value of `lamda`.

Answer

To find the value of \( \lambda \) for which the equation \( x^2 - y^2 - 2x + 2y + \lambda = 0 \) represents a degenerate conic, we will follow these steps:

### Step 1: Identify the coefficients
The given equation can be rewritten in the standard form of a conic:
\[ x^2 - y^2 - 2x + 2y + \lambda = 0 \]
From this, we can identify the coefficients:
- \( A = 1 \) (coefficient of \( x^2 \))
- \( B = -1 \) (coefficient of \( y^2 \))
- \( H = 0 \) (coefficient of \( xy \))
- \( G = -1 \) (coefficient of \( x \))
- \( F = 1 \) (coefficient of \( y \))
- \( C = \lambda \) (constant term)

### Step 2: Set up the discriminant condition
For the conic to be degenerate, the discriminant must be equal to zero. The discriminant \( D \) is given by the formula:
\[ D = A \cdot B - H^2 \]
Substituting the values we found:
\[ D = 1 \cdot (-1) - 0^2 = -1 \]
However, we need to consider the full determinant of the conic which is given by:
\[ D = \begin{vmatrix} A & H & G \\ H & B & F \\ G & F & C \end{vmatrix} \]
Substituting the coefficients:
\[ D = \begin{vmatrix} 1 & 0 & -1 \\ 0 & -1 & 1 \\ -1 & 1 & \lambda \end{vmatrix} \]

### Step 3: Calculate the determinant
Calculating the determinant using the formula for a 3x3 matrix:
\[ D = A(B \cdot C - F \cdot F) - H(G \cdot C - F \cdot H) + G(H \cdot F - B \cdot G) \]
Substituting the values:
\[ D = 1((-1) \cdot \lambda - 1 \cdot 1) - 0 + (-1)(0 - (-1) \cdot (-1)) \]
Simplifying this:
\[ D = -\lambda - 1 - 1 = -\lambda - 2 \]

### Step 4: Set the discriminant to zero
For the conic to be degenerate, we set the determinant equal to zero:
\[ -\lambda - 2 = 0 \]

### Step 5: Solve for \( \lambda \)
Solving the equation:
\[ -\lambda = 2 \implies \lambda = -2 \]

### Conclusion
Thus, the value of \( \lambda \) for which the equation represents a degenerate conic is:
\[ \lambda = -2 \]