the parabola tangent theorem

The Parabola-Tangent Theorem
If P is a parabola and <L> is a family of lines, then
(1) with one exception, each family contains exactly one tangent to P, and
(2) in the exceptional case, each member of the family cuts C once.

This depends on the result for a standard parabola:

The Parabola P0 : y2 = x.

(1) The line y = k cuts P0 exactly once.
(2) The line x = ny+c

  • cuts P0 twice if n2+4c > 0,
  • is a tangent to P0 if n2+4c = 0, and
  • does not meet P0 if n2+4c < 0.
By Theorem AC2, there is an affine transformation s mapping P to P0.
As s is affine <L> maps to the parallel family <s(L)>. By the Parabola
Theorem, the latter family contains exactly one tangent M. As s-1 is also
affine, s-1(M) is the only tangent in <L>.

affine tangents page