unfortunately I cann't offer you a solution.
I've attached a diagram of both surfaces:
x² + y² + z² = 9 describes a sphere with the origin as it's center and the radius = 3
z² = x² + y² describes a double-cone with the origin as it's vertex .
The locus where both surfaces intersect are 2 circles.
To show that these surfaces are orthogonal you have to show that the normal vectors of the surfaces are orthogonal. You are refering to a "condition of orthogonality", so maybe you can use this condition.
Hope this helps a little bit.