設(shè)

P(x1，x12/(2p)）? ??

Q(x2，x22/(2p））? ?

A(√(2pa），a）

B(0，-a）

拋物線

x2=2py

直線PQ

y=kx-a

有

x2=2p(kx-a)

即

x2-2pkx+2pa=0

即

x1+x2=2pk?

x1x2=2pa

k2≥2a/p

有

x12x22

1+(x12+x22)/(4p2)+x12x22/(16p^4)

=

4p2a2

1+(4p2k2-4pa)/(4p2)+a2/(4p2)

≥

4p2a2

1+(8ap-4pa)/(4p2)+a2/(4p2)

=

4p2a2+4a3p+a^4

=

(2pa+a2）2

即

OP2·OQ2≥OA^4

即

OP·OQ≥OA2

有

x12+(x12/(2p）+a)2

x22+(x22/(2p）+a)2

=

4p2a2+4p2a2(4p2k2-4pa）/(4p2)

+4pa3+a2(4p2k2-4pa）

+a^4

+a3(4p2k2-4pa）/p

+4pa3+4a^4

+a3(4p2k2-4pa）/p

+a2((4p2k2-4pa）2-8p2a2）/(4p2)

+a^4

≥

4p2a2+16a3p

+16a^4

=

(2pa+4a2）2

即

BP2·BQ2≥BA^4

即

BP·BQ≥BA2

得證