If a hyperbola has one focus at the origin and its eccentricity is $\sqrt{2}$ . One of the directrices is x + y + 1 = 0. Then the centre of the hyperbola is 
 

Given focus (S)=(0, 0) , $\mathrm{e}=\sqrt{2}$,  directrix is $\mathrm{x}+\mathrm{y}+1=0$

 let P(x, y) be any point on hyperbola then $\frac{\mathrm{SP}}{\mathrm{Pm}}=\mathrm{e}$

$$\begin{gathered} \Rightarrow {\left(\mathrm{SP}\right)}^2=e^2 {\mathrm{Pm}}^2 \\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2=2\frac{{(\mathrm{x}+\mathrm{y}+1)}^2}{2} \\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{x}}^2+{\mathrm{y}}^2+1+2\mathrm{xy}+2\mathrm{x}+2\mathrm{y} \\ \Rightarrow \mathrm{let} \mathrm{S}=2\mathrm{xy}+2\mathrm{x}+2\mathrm{y}+1=0 \\ \frac{\partial S}{\partial x}=0\Rightarrow 2\mathrm{y}+2=0\Rightarrow \mathrm{y}=-1\to \left(1\right) \\ \frac{\partial S}{\partial y}=0\Rightarrow 2\mathrm{x}+2=0\Rightarrow \mathrm{x}=-1\to \left(1\right) \\ \therefore \mathrm{Centre} =(-1, -1). \end{gathered}$$