Find the equation of the hyperbola whose : focus is (1,1) directrix is `3x+4y+8=0` and eccentricity `=2`

Explanation

Let `P(x,y)` be a point on the hyperbola.

And we know that,

Distance of point `P` from focus`=` Eccentricity `xx` Distance of point `P` from directrix.

`therefore sqrt{(x-1)^2+(y-1)^2}=2({3x+4y+8}/sqrt{9+16})`

`implies (x-1)^2+(y-1)^2=4({3x+4y+8}/5)^2`

`implies x^2+1-2x+y^2+1-2y=4/25(9x^2+16y^2+24xy+64+48x+64y)`

`implies 25x^2+25y^2-50x-50y+50=36x^2+64y^2+96xy+192x+256y+256=0`

`implies 11x^2+39y^2+96xy+242x+306y+206=0`

...