Two surfaces are said to be orthogonal at a point P if the normals to their tangent planes are perpendicular at P. Show that the surfaces z= 1/2(x²+y² -1) and z=1/2(1- x²-y²) are orthogonal at all points of intersection.
Here's an outline - details left for you.
Get a normal to the tangent plane for each surface: A normal to a tangent plane is given by (z_x)i + (z_y)j - k.
Find the curve of intersection of the two surfaces by solving $\displaystyle x^2 + y^2 - 1 = 1 - x^2 - y^2$: $\displaystyle x^2 + y^2 = 1$
Take the dot product of the two normals and substitute $\displaystyle x^2 + y^2 = 1$ into the result. The resulting expression should simplify to zero.