18.2 Analytical Geometry

The following exercise is cited from a German textbook.

The plane E and the set of planes gp mit p ∈ ℝ are given.

   (   )                   (   )     (        )
     1                       0            2
E :( 2 ) ⋅⃗x =  18  gp : ⃗x = ( 0 ) + λ (   p   )
     2                       9         - p - 1

  1. Find an equation of the sphere with midpoint P(2|1|2) that has E as tangent-plane. Compute the point where E touches the shpere.
  2. Show that gp lies in E for all values of p.
  3. Which lines are tangents of the sphere from part a)?

18.2.1 Solution with Archimedes

  1. The sphere has the equation (     (  ) )
        2
( ⃗x - ( 1) )
        22 = (   )
 3 132.

    PIC

  2. The claim can, of course, only be verified, not proven with Archimedes.
  3. We can find out by trying that the solution for the parameter p is 2, or we could construct the line (as the parameter is not asked for) and find out the equation by right click - description.