4 圓錐曲線中的定點(diǎn)定值問題(教師版)

1、聯(lián)邦理科高二寒假1第四講圓錐曲線中的定點(diǎn)定值問題一、直線恒過定點(diǎn)問題例1.已知?jiǎng)狱c(diǎn)在直線上，過點(diǎn)分別作曲線的切線，切點(diǎn)E:2ly??E2:4Cxy?EAEB為、求證：直線恒過一定點(diǎn)，并求出該定點(diǎn)的坐標(biāo)；ABAB解：設(shè)，)2(?aE)4()4(222211xxBxxAxyxy2142????)(2141121兩兩兩兩兩兩兩兩兩兩兩兩兩兩兩兩ExxxxyA????整理得：)(21421121xaxx?????082121???axx同理可得

2、：222280xax???820822121221??????????xxaxxaxxxx兩兩兩兩兩兩，又)24(2?aaAB兩兩兩兩兩2212121212124442ABxxyyxxakxxxx?????????，.2(2)()22aaAByxa?????直線的方程為2()2ayxAB???即過定點(diǎn)02例2、已知點(diǎn)00()Pxy是橢圓22:12xEy??上任意一點(diǎn)，直線l的方程為0012xxyy??，直線0l過P點(diǎn)與直線l垂直，點(diǎn)M（

3、1，0）關(guān)于直線0l的對(duì)稱點(diǎn)為N，直線PN恒過一定點(diǎn)G，求點(diǎn)G的坐標(biāo)。解：直線0l的方程為0000()2()xyyyxx???，即000020yxxyxy???設(shè))01(?M關(guān)于直線0l的對(duì)稱點(diǎn)N的坐標(biāo)為()Nmn則0000001212022xnmyxnmyxy???????????????，解得320002043200002002344424482(4)xxxmxxxxxnyx??????????????????直線PN的斜率為432

4、0000032000042882(34)nyxxxxkmxyxx???????????聯(lián)邦理科高二寒假3設(shè)()BBBxy則2222(2)222122Bkkkkxkk????????同理可得222222Akkxk????，則2422ABkxxk???28(1)(1)2ABABkyykxkxk????????所以直線AB的斜率2ABABAByykxx????為定值。例4、已知?jiǎng)又本€與橢圓相交于、兩(1)ykx??22:1553xyC??AB

5、點(diǎn)已知點(diǎn)，求證：為定值.7(0)3M?MAMB?????????解:將代入中得(1)ykx??221553xy??2222(13)6350kxkxk?????，4222364(31)(35)48200kkkk?????????，2122631kxxk????21223531kxxk???所以112212127777()()()()3333MAMBxyxyxxyy????????????????2121277()()(1)(1)33xxk