Are you taking Calculus? This is a pretty standard Calculus problem. As you say "Two surfaces are said to intersect orthogonally at a point P if their
tangent planes are perpendicular at the point P." Are you saying you do not know how to find the tangent planes? That is also a pretty basic Calculus problem- similar to "find the tangent line to a curve" that is pretty much the first thing you learn in Calculus!
But even simpler is the fact that two planes are perpendicular (and so the surfaces are perpendicular) if and only if their normal vectors are perpendicular. What are the normal vectors to and at (1, 1, 1)?