klein view page

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.

Proof
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

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.
Proof 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>.