If AB slope does not exist, then AB is x = p.

y ^ 2 = 4p ^ 2。

y=2p，-2p

So m = 2p, n = |-2p | = 2p.

Then1/m+1/n =1/p.

If the slope exists

Y-0=k(x-p)

y=kx-kp

replace

k^2x^2-(4p+2k^2p)x+k^2p^2=0

x 1+x2=(4p+2k^2p)/k^2

x 1x2=p^2

Line x=-p

Defined by parabola, the distance from the point on the parabola to the focus is equal to the distance from the directrix.

So m = AF = distance to the alignment line = x1-(-p) = x1+p.

Similarly, n = x2+p.

m+n=x 1+x2+2p=(4p+2k^2p)/k^2+2p=(4p+4k^2p)/k^2

mn=(x 1+p)(x2+p)=x 1x2+p(x 1+x2)+p^2=(4p^2+4p^2k^2)/k^2

So 1/m+ 1/n=(m+n)/mn.

=(4p+4k^2p)/(4p^2+4p^2k^2)

= 1/p

When the slope does not exist, it is also equal to1/p.

So1/m+1/n =1/p, which is a constant value.