If the equation $x^2-y^2-2x+2y+\lambda =0$ represents a degenerate conic, find the value of $\lambda$.

 

The given conic section is $x^2-y^2-2x+2y+\lambda =0$.

We know that the general conic section equation is $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$.

Comparing both, we get

$$\begin{gathered} a=1 \\ h=0 \\ b=-1 \\ g=-1 \\ f=1 \\ c=\lambda \end{gathered}$$

A degenerate conic is a conic which can be factorized as the product of two linear polynomials.

For a degenerate conic, $∆=0$.

We know that,

$∆=\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|$

Therefore,

$$\begin{gathered} ∆=\left|\begin{array}{ccc}1& 0& -1\\ 0& -1& 1\\ -1& 1& \lambda \end{array}\right| \\ \Rightarrow ∆=1\left[-1\left(\lambda \right)-1\right]-0+(-1)\left[1\left(0\right)-1\right] \\ ∆=0\Rightarrow -\lambda -1+1=0 \\ \Rightarrow \lambda =0 \end{gathered}$$

Therefore, the correct option is $2$.