Find the equation of the hyperbola whose : focus is (2,-1) directrix is `2x+3y=1` 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-2)^2+(y+1)^2}=2({2x+3y-1}/sqrt{4+9})`

`implies (x-2)^2+(y+1)^2=4({2x+3y-1}/sqrt13)^2`

`implies x^2+4-4x+y^2+1+2y=4/13(4x^2+9y^2+12xy+1-4x-6y)`

`implies13 x^2+52-52x+13y^2+13+26y =16x^2+36y^2+48xy-16x-24y+4=0`

`implies 3x^2+23y^2+48xy+36x-50y-61=0`

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