Find the equation of the hyperbola whose : focus is (0,3) directrix is `x+y-1=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-0)^2+(y-3)^2}=2({x+y-1}/sqrt2)`

`implies (x-0)^2+(y-3)^2=4({x+y-1}/sqrt2)^2`

`implies x^2+y^2+9-6y=2(x^2+y^2+1+2xy-2y-2x)`

`implies x^2+y^2+4xy-4x+2y -7=0`

This is the required equation of the hyperbola.