Find the equation of the hyperbola whose focus is (1,1), eccentricity is 2 and equation of directrix is x+y+1=0.

Explanation

Let the focus be S(1, 1) and PM is the length of perpendicular from point P(x,y) the length of perpendicular from point P(x,y) to the directrix where P lies on the hyperbola.
From the definition of hyperbola
`PS=e*PM`
`sqrt((x-1)^(2)+(y-1)^(2))=(2(x+y+1))/(sqrt(1^(2)+1^(2)))`
`rArr`
`rArr" "[(x-1)^(2)+(y-1)^(2)]=2(x+y+1)^(2)`
`rArr" "(x^(2)-2x+1+y^(2)-2y+1)=2(x^(2)+y^(2)+1+2xy+2x+2y)`
`rArr" "x^(2)+y^(2)+4xy+6x+6y+1=0.`