Diameter of a conic is the locus of middle points of a series of its parallel chords.

Let the parabola be `y^2 = 4ax` ....... (1)

Let `AB` be one of the chords of a series of parallel chords having slope `m`.

Let `P(alpha, beta)` be the middle point of chord `AB`, then equation of `AB` will be `T= S_1`

or `y beta - 2a (x +a) = beta^2 - 4 alpha a`................(2)

Slope of line `(2) = (2a)/m`

but slope of line `(1)` i.e. line `AB` is `m`.

`:. (2a)/beta =m ` or `beta =(2a)/m`

Hence locus of `P(a, beta)` i.e. equation of diameter

(which is the locus of a series of a parallel chords having slope `m`) is

`y = (2a)/m` .......... (3)

Clearly line `(3)` is parallel to the axis of the parabola. Thus a diameter of a parabola is parallel to its axis.