Parabola

for a parabola,
if normal at t cuts the parabola again at t₂, then

t₂ = -t -2t. (this can be proved..)

as t is -ve.
-t and -2/t both are +ve..
thus we can apply A.M. ≥ G.M.

thus, (-t)+(-2/t)2 ≥ √(-t).(-2/t)
thus, t₂ ≥ 2√2
least value of t₂ = 2√2.
thus, t= -√2.

y² =4 .52 x
thus, a= 5/2
thus P = (at²,2at) = (5, -5√2)
thus, Q = (at₂²,2at₂) = (20, 10√2)
thus, PQ = 15√3.
i am getting this as answer...

Thank you very much.

it can also be done by maxima and minima...