发布网友 发布时间:2024-05-08 19:38
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热心网友 时间:2024-05-10 17:39
S2^2=4S1S3成立,证明如下: 证:设M(x1,y1),N(x2,y2) 则由抛物线的定义得 |MM1|=|MF|= x1+p/2,|NN1|=|NF|= x2+p/2, 于是 S1= 1/2|MM1||F1M1|= 1/2(x1+p/2)|y1|, S2= 1/2|M1N2||FF1|= 1/2p|y1-y2|, S3= 1/2|NN1||F1N1|= 1/2(x2+p/2)|y2|, ∵S2^2=4S1S3 互推 (1/2p|y1-y2|^2=4×1/2(x1+p/2)|y1|• 1/2(x2+p/2)|y2| 互推 [1/4p^2(y1+y2)^2-4y1y2]= [x1x2+p/2(x1+x2)+p^2/4]|y1y2|, 将 {x1=my1+p/2 x2=my2+p/2与 {x1+y2=2mp y1y2=-p^2代入上式化简可得 p^2(m^2p^2+p^2)=p^2(m^2p^2+p^2),此式恒成立. 故S2^2=4S1S3成立.