Mathematics

Problem

75. Focal Chords A focal chord of a parabola is a chord of the parabola that passes through the focus. (a) Prove that the

x

-coordinates of the endpoints of a focal chord of

x^{2}=4 p y

are

x=2 p\left(m \pm \sqrt{m^{2}+1}\right)

, where

m

is the slope of the focal chord. (b) Using part (a), prove that the minimum length of a focal chord is the focal width

|4 p|