已知橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830477318.png)
:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830493766.png)
(
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830508498.png)
)過點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830524516.png)
,其左、右焦點(diǎn)分別為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830539443.png)
,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830571660.png)
.
(1)求橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830477318.png)
的方程;
(2)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830602550.png)
是直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830617357.png)
上的兩個動點(diǎn),且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830633742.png)
,則以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830649513.png)
為直徑的圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830664313.png)
是否過定點(diǎn)?請說明理由.
試題分析:(1)設(shè)點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830758441.png)
的坐標(biāo)分別為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830773639.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830789920.png)
,故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240108308051189.png)
,可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830820339.png)
,
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240108308361816.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830898499.png)
,
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830914808.png)
,所以橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830477318.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830695748.png)
.
(2)設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830602550.png)
的坐標(biāo)分別為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830976656.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830992803.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010831023731.png)
. 由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010831039776.png)
,可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240108310541066.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010831070506.png)
,
又圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830664313.png)
的圓心為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010831085740.png)
半徑為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010831101579.png)
,故圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830664313.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240108311321292.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240108311481079.png)
,也就是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240108311631076.png)
,令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010831210391.png)
,可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010831226397.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010831226291.png)
,
故圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830664313.png)
必過定點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830711477.png)
和
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824010830727480.png)
.
點(diǎn)評:第一小題利用向量的坐標(biāo)運(yùn)算及橢圓定義可求得方程;第二小題判定曲線是否過定點(diǎn)只需看曲線方程中能否轉(zhuǎn)化出與參數(shù)無關(guān)的關(guān)系式
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