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

$$\begin{gathered} Given focus=(0,0),e=\sqrt{2} \\ and equation of directrix is x+y+1=0 \\ Let P(x,y) be any point on hyperbola then \\ SP=ePM \\ \Rightarrow S{P}^2=e^{2}P{M}^2 \\ \Rightarrow x^2+y^2=2\frac{{(x+y+1)}^2}{2} \\ \Rightarrow Hyperbola is S\equiv 2xy+2x+2y+1=0 \\ Now \frac{\partial S}{\partial x}=0\Rightarrow 2y+2=0\Rightarrow y=-1 and \\ \frac{\partial S}{\partial y}=0\Rightarrow 2x+2=0\Rightarrow x=-1 \\ So,centre=(-1,-1) \end{gathered}$$