what does tangent plane mean
The tangent plane at the point x=a, y=b is the plane that contains all the tangent vectors (at the point) of curves on the surface that pass through the point (a,b,f(a,b)). Another way to define it is to say that this is the plane through the point (a,b,f(a,b)) which is perpendicular to the NORMAL to the surface at that point. The NORMAL to the surface at a point is just defined as the direction perpendicular to all tangent vectors to the surface at that point. Therefore all we need to do to find the equation of the tangent plane is to find a normal to the surface at that point.