已知橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958378318.png)
的兩個焦點分別為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958393452.png)
和
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958409432.png)
,離心率
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958424516.png)
.
(1)求橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958378318.png)
的方程;
(2)若直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958456658.png)
(
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958456418.png)
)與橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958378318.png)
交于不同的兩點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958487300.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958502309.png)
,且線段
的垂直平分線過定點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958534608.png)
,求實數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958549312.png)
的取值范圍.
(1)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958565654.png)
;(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240449585801116.png)
.
試題分析:(1)求橢圓的標準方程
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958596766.png)
,要找兩個等式以確定
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958612396.png)
,本題中有焦點為,說明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958627304.png)
,又有離心率,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958643647.png)
,由此再加上
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958658549.png)
可得結論;(2)直線與圓錐曲線相交問題,又涉及到交點弦,因此我們都是把直線方程(或設出)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958674598.png)
與橢圓方程聯(lián)立方程組,然后消去
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958690310.png)
(有時也可消去
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958705266.png)
)得關于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958705266.png)
(或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958690310.png)
)的一元二次方程,再設交點為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958814415.png)
坐標為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958814728.png)
,則可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958846429.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958861397.png)
,(用
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958877448.png)
表示),于是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958518396.png)
中點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958908315.png)
坐標
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958939564.png)
可得,其中
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958939632.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958955682.png)
,而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958970553.png)
,從而建立了
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958877448.png)
的一個等量關系,在剛才的一元二次方程中,還有判別式
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959002426.png)
,合起來可得出關于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958549312.png)
的不等式,從而求出其范圍.
試題解析:(1)由已知橢圓的焦點在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958705266.png)
軸上,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958627304.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959064574.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959080195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959095434.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959095352.png)
, 2分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959080195.png)
橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958378318.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958565654.png)
4分
(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240449591581079.png)
,消去
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958690310.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240449591891080.png)
6分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959204235.png)
直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959220280.png)
與橢圓有兩個交點,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959080195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959236344.png)
,可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959267647.png)
(*) 8分
設
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959282616.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959314644.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959080195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959360916.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959080195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958518396.png)
中點的橫坐標
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959423801.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958518396.png)
中點的縱坐標
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959454975.png)
10分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959080195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958518396.png)
的中點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959501970.png)
設
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958518396.png)
中垂線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959548307.png)
的方程為:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959563760.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959204235.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958908315.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959548307.png)
上,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959080195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044958908315.png)
點坐標代入
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959548307.png)
的方程可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959782722.png)
(**) 12分
將
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959267647.png)
(*)代入解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959813601.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959828587.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044959080195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240449585801116.png)
14分
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