About the Cartesian equation?

Question

Let P be the tangent plane to the surface xyz=a^3 at the point (r,s,t). Find the Cartesian equation of P.
Let T be the tetrahedron (1/3*(area of base x height)) formed by the coordinate planes, x=0, y=0, z=0, and the tangent plane P.
Show that the volume of T is independent of the point (r,s,t).

Answer 1

Let F(x,y,z) = xyz - a³. Then the gradient of F, evaluated at (r,s,t), is a vector normal to the surface at (r,s,t). Let this vector be called N. Let (x,y,z) be a point in the tangent plane P distinct from (r,s,t). Then the vector

v = (x-r)i + (y-s)j + (z-t)k

lies in P, and hence is perpendicular to N. Therefore, v•N=0. This is an equation for P.

Note that, since (r,s,t) is on the surface, rst = a³. Using this, you will find that the equation for P may be written

(x/r) + (y/s) + (z/t) = 3

The second question is easy if you draw a picture of the tetrahedron. Note where P intersects each of the coordinate axes. You will find that the volume of T is (9/2)a³ no matter what (r,s,t) is.