$|AP-BP|=6$
=>$AP-BP=\pm 6$
=>$ AP=BP\pm 6$
=> $ \sqrt {x^2+(y-4)^2}=\sqrt {x^2+(y+4)^2} \pm 6$
=> $x^2+y^2-8y+16=x^2+y^2+8y+16+36 \pm 12 \sqrt {x^2+(y+4)^2}$
$-16y-36=\pm 12 \sqrt {x^2+(y+4)^2}$
=>$ 4y+9=\pm 3 \sqrt {x^2+(y+4)^2}$
=> $ 16y^2+72y+81=9(x^2+y^2+8y+16)$
=>$7y^2-9x^2=63$
=> $\large\frac{y^2}{9} -\frac{x^2}{7}$$=1$
Hence d is the correct answer.