Let Γ be a parabola with focus F and directrix d.
A line through F intersects Γ in points P₁ and P₂.
The feet of the perpendiculars from P₁ and P₂ on
d are Q₁ and Q₂ respectively.
The midpoint of line segment Q₁Q₂ is Q₀.

Prove that the rays Q₀P₁ and Q₀P₂ are orthogonal
and that they are tangent to Γ.

without loss of generality:
let the vertex be at the origin,
let the directrix be y=-p
let the focus be (0,p)
then the parabola is given by
x^2=4py

now let P1 be (2pk,pk^2) for some real number k
then the line P1-F is given by
y-pk^2=(pk^2-p)(x-2pk)/(2pk)
solving for the other intersection gives
P2 at (-2p/k,p/k^2)

thus the feet Q1,Q2 are
Q1: (2pk,-p)
Q2: (-2p/k,-p)
thus
Q0: (p(k^2-1)/k,-p)
thus the rays Q0P1 and Q0P2 have slopes
Q0P1: m1=k
Q0P2: m2=-1/k
m1*m2=k*-1/k=-1
thus Q0P1 and Q0P2 are orthogonal

Posted by Daniel on 2011-12-01 10:35:19