The directrix of a hyperbola is $x-y+3=0$. Its focus is at $(-1,1)$ and has an eccentricity of $3$. I am trying to find the equation of the hyperbola.
I tried using the fact that $\frac{\mathrm{PF}}{\mathrm{PD}} = \mathrm{eccentricity}$, where PF is distance of any point P on the hyperbola to the focus, and PD is the distance of P to the directrix. I applied distance of a point from a line and distance between two points formulae and substituted the values in the above equation. With that, I got $$17x^2+17y^2+2xy+30x+42y+27=0.$$
But the answer given is: $7x^2+7y^2-18xy+50x-50y+77=0$.
Is my approach to this problem correct?
Your approach is correct, but you made a mistake in computing the ratio. In particular, instead of simplifying $$\frac{(x+1)^2 + (y-1)^2}{(x-y+3)^2/2} = 9,$$ you simplified $$\frac{(x-y+3)^2/2}{(x+1)^2 + (y-1)^2} = 9.$$