Hi,
this is the shortest i cud think of
T = 0 is the eqn of chord of contact
suppose y^2 = 4ax
and T is (j,k)
so T = 0
implies
y = (2a/k)x + (j/k)
Here use parameric eqn of parabola
let P be (at^2,2at)
so slope of tangent is 1/t
as PQ is normal , it's slope is -t
so -t = 2a/k
so k = -2a/t
now
(k - 2at)/(h - at^2) = 1/t ..slope of normal
using value of k we get h to be = -a(t^2+2)
let x +a = 0 divide PT in ratio U:1 at point F ....U:1 = TF:FP
so
[U(at^2) + ( - a)(t^2 + 2) ]/(U+1) = -a

so u get
U:1 = 1:1
i believe that in case u have to prove something find some ratio...always use the parametric equation , unless the nmerical values of coordinates is known.

U become a better judge urself as u practice more !!