已知點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010733880302.png)
是橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010733896964.png)
的右焦點,點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010733896674.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010733927610.png)
分別是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010733943266.png)
軸、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010733958310.png)
軸上的動點,且滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010733974729.png)
.若點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010733989289.png)
滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734005845.png)
.
(Ⅰ)求點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010733989289.png)
的軌跡
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734036313.png)
的方程;
(Ⅱ)設(shè)過點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010733880302.png)
任作一直線與點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010733989289.png)
的軌跡交于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734083300.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734083309.png)
兩點,直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734099376.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734130370.png)
與直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734130378.png)
分別交
于點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734145321.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734239304.png)
(
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734255292.png)
為坐標原點),試判斷
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734411530.png)
是否為定值?若是,求出這個定值;若不是,
請說明理由.
(Ⅰ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734426582.png)
(Ⅱ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734411530.png)
的值是定值,且定值為
試題分析:(Ⅰ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734457235.png)
橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010733896964.png)
右焦點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010733880302.png)
的坐標為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734504468.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734535743.png)
.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734567861.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734582195.png)
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010733974729.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734613633.png)
.
設(shè)點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010733989289.png)
的坐標為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734660500.png)
,由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734005845.png)
,有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240107349101013.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734925904.png)
代入
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734613633.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734426582.png)
.
(Ⅱ)解法一:設(shè)直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734988396.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735003534.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735035789.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735050794.png)
,
則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735066816.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735081842.png)
.
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735113922.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735128830.png)
, 同理得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735144822.png)
.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240107351751033.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735191997.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240107352061141.png)
.
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735222947.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735253733.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735269641.png)
.
則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240107352841395.png)
.
因此,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734411530.png)
的值是定值,且定值為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735331262.png)
.
解法二:①當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735347473.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735378587.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735393604.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735409646.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735596650.png)
.
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735612753.png)
得點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734145321.png)
的坐標為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735643623.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735659741.png)
.
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735690767.png)
得點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734239304.png)
的坐標為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735721580.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735721696.png)
.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240107357521156.png)
.
②當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734988396.png)
不垂直
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010733943266.png)
軸時,設(shè)直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734988396.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735799814.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735815796.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735830807.png)
,同解
法一,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240107352061141.png)
.
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240107358611038.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735877785.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735269641.png)
.
則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240107352841395.png)
.
因此,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010734411530.png)
的值是定值,且定值為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010735331262.png)
.
點評:解決此類題目的關(guān)鍵是熟練掌握求軌跡方程的方法(消參法),以及設(shè)點利用點表示
有關(guān)的向量的表達式即可,此題對計算能力要求較高.
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