Since the chords are all parallel, they all make the same angle with the axis of x. Let the tangent of this angle be on.
The equation to QR, anyone of these chords, is therefore
y = mx + c ........(1)
where c is different for the several chords, but m is the same.
This straight line meets the parabola y2 = 4ax in points whose ordinates are given by
Let the roots of this equation, i.e. the ordinates of Q and R, be y' and y" and let the coordinates of V, the middle point of QR, be (h, k).
from equation (2).
The coordinates of V therefore satisfy the equation
y = 2a/m,
so that the locus of V is a straight line parallel to the axis of the curve,
The straight line y = 2a/m meets the curve in a point P, whose ordinate is 2a/m and whose abscissa is therefore a/m2.
The tangent at this point is,
and is therefore parallel to each of the given chords.
Hence the locus of the middle points of a system of parallel chords of a parabola is a straight line which is parallel to the axis and meets the curve at a point the tangent at which is parallel to the given system.
