Answer : $7x^2+2xy+7y^2+10x-10y+7=0$
Given focus is $(1,-1)$ and equation of the directrix is $x-y=3$
Eccentricity $e=\large\frac{1}{2}$
Hence $SP=ePM$
$SP^2=e^2PM^2$
(ie) $SP^2=\large\frac{1}{4}$$(PM)^2$
$4SP^2=PM^2$
(ie) $4[(x+1)^2+(y-1)^2]=\big(\large\frac{x-y-3}{\sqrt{1^2+(-1)^2}}\big)^2$
$\Rightarrow 8(x^2+y^2+2x-2y+2)=(x-y+3)^2$
On simplifying we get,
$7x^2+2xy+7y^2+10x-10y+7=0$