The equation of the conic with focus at $(1,-1)$, directrix along $x-y+1=0$ and with eccentricity $\sqrt{2}$ is:

Here, focus (S) = ($\mathrm{I},-1$), eccentricity ($e$) =.$\sqrt{2}$ From definition, $SP=ePM$

$\sqrt{(x-1{)}^2+(y+1{)}^2}=\frac{\sqrt{2}\cdot (x-y+1)}{\sqrt{1^2+1^2}}$

$\begin{array}{l}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}(x-1{)}^2+(y+1{)}^2\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=(x-y+1{)}^2\end{array}$

$\Rightarrow$ $2xy-4x+4y+1=0$, which is the required equation of conic (Rectangular hyperbola)