抛物线与直线关系

l方程： x/a+y/b=1
x=a-ay/b
代人y^2=2px得：
y^2=2pa-2pay/b
y^2+2pay/b-2pa=0
y1+y2=-2pa/b,y1y2=-2pa
所以，
1/y1+1/y2=(y1+y2)/y1y2
=(-2pa/b)/(-2pa)
=1/b

tan∠MON=(Kmo+Kno)/(1-Kmo*Kno)
=(y1/x1+y2/x2)/(1-y1y2/x1x2)
=(2p/y1+2p/y2)/(1-y1y2/x1x2)
=2p(1/y1+1/y2)/(1-y1y2/x1x2)
=2p/b(1-y1y2/x1x2)
=2p/b(1-y1y2/(y1^2*y2^2/4p^2))
=2p/b(1-4p^2/y1y2)
=2p/b(1-4p^2/(-2pa))
=2p/b(1+2p/a)
=2p/b(1+2p/2p)
=2p/2b
=p/b

(1)设直线l的方程截距式为x/a+y/b=1,代入抛物线方程，消去x，得
by²+2pay-2pab=0
y1+y2=-2pa/b
y1y2=-2pa
1/y1+1/y2=(y1+y2)/y1y2=-2pa/b÷(-2pa)=1/b

(2)设l的方程，x-2p=my → (x-my)/2p=1，代入抛物线方程
y²=2px·(x-my)/2p
y²+mxy-x²=0，x≠0，两边同时除以x²，得
(y/x)²+m(y/x)-1=0
y1y2/x1x2=-1
向量OM·向量ON=x1x2+y1y2=0
OM⊥ON，
∠MON=90°