(1)F(0,p/2)
設(shè)直線l：y=kx+p/2,A(x1,y1),B(x2,y2)
聯(lián)立x^2=2py
y=kx+p/2
x^2-2kpx-p^2=0
x1+x2=2kp
x1x2=-p^2
|AB|=√(1+k^2)|x1-x2|
=√(1+k^2)×2p√(1+k^2)
=2p(1+k^2)
d=|p/2|/√(1+k^2)
S△AOB=1/2×|AB|×d
=p^2/2√(1+k^2)≥p^2/2
S△AOBmin=p^2/2=8
解得p=4
x^2=8x
(2)設(shè)M(a,b)
y=1/8x^2
y'=1/4x
切線斜率k=1/4x1
y-y1=1/4x1(x-x1)
y=1/4x1(x-x1)+y1
=1/4x1(x-x1)+1/8x1^2
=1/4x1x-1/8x1^2
令x=0,
y=-1/8x1^2
即N(0,-1/8x1^2)
向量FA=(x1,y1-p/2)=(x1,1/8x1^2-p/2)
向量FN=(0,-1/8x1^2-p/2)
向量FM=(a,b-p/2)=(x1,-p)
所以a=x1,b=-p/2 =-2
所以M在y=-2上,
即M在準(zhǔn)線上