The equation of the conic with focus at (1,-1) directrix along `x-y+1=0` and eccentricity `sqrt(2)` is a.`x y=1` b. `2x y+4x-4y-1=0` c. `x^2-y^2=1` d. `2x y-4x+4y+1=0`

Explanation

We have given focus at (1,-1) directrix along `x-y+1=0`
and eccentricity `sqrt(2)`of a conic.

Using Definition of hyperbola
`PS^2=e^2⋅PM^2`

`(x−1)^2+(y+1)^2=2((x−y+1)/sqrt2)^2`

`​(x^2+y^2−2x+2y+2)=(x^2+y^2+1−2xy+2x−2y)`

`implies2xy−4x+4y+1=0`

Hence, option (d) is correct.