(本小題滿分12分)
如圖,A、B分別是橢圓
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的公共左右頂點,P、Q分別位于橢圓和雙曲線上且不同于A、B的兩點,設直線AP、BP、AQ、BQ的斜率分別為k
1、k
2、k
3、k
4且k
1+k
2+k
3+k
4=0。
(1)求證:O、P、Q三點共線;(O為坐標原點)
(2)設F
1、F
2分別是橢圓和雙曲線的右焦點,已知PF
1//QF
2,求
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的值。
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(1)設
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,則
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
………………2分
又
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所以
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
…………4分
由
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即
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所以O、P、Q三點共線 ………………6分
(2)
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由PF
1//QF
2知|OP|:|OQ|=
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因為O、P、Q三點共線,
所以
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…………① …………7分
設直線PQ的斜率為k,則

…………②
由①②得
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………………
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10分
又


………………12分
從而
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6…………14分
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