Tangent planes containing a line

Find the tangent planes to the surface (sphere) containing line r (through P(19, 9, 1), direction (2, 2, 1)).

Step-by-step solution

If the line cuts the surface, no tangent plane containing it can avoid cutting it.
If it is outside, two tangent planes come from it (sphere) or one (cone, depending on the position).
Find the point of tangency and check that the line lies in the plane.
Check that the plane touches the surface at a single point or along a generator.

Result: 2 tangent plane(s). Dos planos, uno por cada rama del cono tangente.

A line outside a sphere has two tangent planes; a secant line has none.