Diameter

The locus of the mid-points of a system of parallel chords of a conic is known as its diameter.

Equation of diameter of a parabola:

Let y = mx + c be a system of parallel chords of the parabola y² = 4ax.

Here, m is a constant and c is a variable.

The line y = mx + c meets the parabola y² = 4ax in the points say P and Q whose ordinates (say y₁ and y₂) are the roots of the equation.

∴ 

my² - 4ay + 4ac = 0.

∴ y₁ + y₂ = 4a/m.

Let M (h, k) be the mid-points of PQ. Then,

k = 2a/m.

Hence, the locus of (h, k) is y = 2a/m.