18.2 Analytical Geometry

The following exercise is cited from a German textbook.

The plane E and the set of planes g_p mit p are given.

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 g_p 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. • Enter the plane via ’Typical tasks - Plane - General form’. One has to know that the given normal form is the same as x + 2y + 2z = 18.
   • Enter the point by double-right-clicking into the scene an setting the co-ordinates afterwards.
   • Drop the perpendicular from P to E, name the perpendicular L.
   • Measure the distance from P to L. You can draw a vector or use the Macro ’Distance - PointPoint’.
   • Draw the sphere around P through L.

   The sphere has the equation ² = ².

   

2. The claim can, of course, only be verified, not proven with Archimedes.
   • Generate a slider via ’Extras - Slider’, name it ’pp’ (as the name P is already used, there mustn’t be a second object of that name, even if you use the lowercase letter).
   • Enter the equation of the line at ’Typical tasks - Line in parametric form’ with the parameter pp like it is written in the exercise. Replace ’p’ with ’pp’. Select the checkbox ’Expression contains terms’.
   • If you change the value of pp with the slider you will see that the line moves within E.
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.