Prove that the semi-latus rectum of the parabola y^(2) = 4ax` is the harmonic mean between the segments of any focal chord of the parabola.

Explanation

Let PQ be focal chord of a given parabola `y^(2) = 4ax`
Let `P(at^(2), 2at)`. Since P, S, Q are collinear, hence coordinates of Q are given by `((a)/(t^(2)),-(2a)/(t))`
Let `l_(1) and l_(2)` be the sections of the focal chord.
Then,
`l_(1) = SP = PM=a+at^(2) = a(1+t^(2))`
`l_(2) = SQ = QN = a+(a)/(t^(2)) = (a(1+t^(2)))/(t^(2))`
Now Harmonic mean of `l_(1)` and `l_(2)`
`=(2l_(1)l_(2))/(l_(1)+l_(2))=(2)/((1)/(l_(1))+(1)/(l_(2)))=(2)/((1)/(a(1+t^(2)))+(t^(2))/(a(1+t^(2))))`
`=2a =` semi-latus rectum.