抛物线与直线关系如图,o为坐标原点,直线l在x轴和y轴上的截距分别是a和b（a＞0,b不等于0）,且交抛物线y2=2px(p＞0)于M(x1,y1)N(x2,y2)两点.证明1/y1+1/y2=1/b;当a=2p时,求∠MON的大小.

Question

抛物线与直线关系
如图,o为坐标原点,直线l在x轴和y轴上的截距分别是a和b（a＞0,b不等于0）,且交抛物线y2=2px(p＞0)于M(x1,y1)N(x2,y2)两点.证明1/y1+1/y2=1/b;当a=2p时,求∠MON的大小.

Answer 1

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