43. Plot the level surface (x^2) -  (y^2) +  z   =  4 along with its tangent plane and normal line at the point (1, 1, 4).

Gradient:

F  =  (x^2)  -  (y^2)  +  z;
grad[x_, y_, z_] =  {D[F, x], D[F, y], D[F, z]}

Tagent  Plane:

p  =  {1, 1, 4};

n  =  grad[1, 1, 4]

n . {x, y, z}  ==  n . p

Normal  Line:

r[t_]  =  p  +  t n

graph  =  
      Plot3D[ 4  -  (x^2) +  (y^2),  {x, -1, 2},  {y, -1, 2}, 
        PlotPoints -> 30]

tplane = 
      Plot3D[4  -  2  x  +  2  y, {x, 0.25, 1.75}, {y, 0.25, 
          1.75}]

nline =  ParametricPlot3D[r[t],  {t, -1, 1}]

Show[graph, tplane, nline, ViewPoint -> {2, -1/2,  1}, 
      BoxRatios -> Automatic]