Let $P(x,y)$ be any point on the hyperbola
$\therefore \large\frac{\text{Distance of P from focus}}{\text{Distance from directrix}}$$=e$
=> $(x-2)^2+(y-2)^2=2^2 \bigg[ \large\frac{x+y-9}{\sqrt 2}\bigg]^2$
$x^2+4-4xy+y^2+4 -4y=4 \bigg[ \large\frac {x+y-81}{2}\bigg]$
$x^2+y^2-4x-4y+8= 2 [x^2+y^2+81+2xy-18y-18x]$
$x^2+y^2-4x-4y+8- 2x^2-2y^2-162-4xy+36y+36x=0$
$-x^2-y^2-4xy+32y+32x-154=0$
$x^2+y^2+4xy-32x-32y+154=0$
Hence A is the correct answer.