$P(2,-2,1),Q(1,2,2),R(0,-3,0),A(x,y,z)$ isn't the origin,distance between A and P,Q,R is the same, what's $\frac{x^2+xy-yz+z^2}{-x^2-y^2+z^2}$

Question

Question:$P(2,-2,1),Q(1,2,2),R(0,-3,0),A(x,y,z)$ is not the origin, the distance between A and P,Q,R is the same, compute the value of$\frac{x^2+xy-yz+z^2}{-x^2-y^2+z^2}$

I can use $PA=QA=RA$ to get three equations and solve for $x,y,z$. But there must be a better way, I need some help. Thanks.

Answer 1

The better way: Find the plane through $P$, $Q$, and $R$. The points $(x,y,z)$ equidistant from $P,Q,R$ are the line through the origin perpendicular to that plane.