如圖,橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449692313.png)
的中心為原點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449707292.png)
,長軸在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449723266.png)
軸上,離心率
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449738516.png)
,又橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449692313.png)
上的任一點到橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449692313.png)
的兩焦點的距離之和為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449770306.png)
.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240434498166640.jpg)
(1)求橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449692313.png)
的標(biāo)準(zhǔn)方程;
(2)若平行于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449848310.png)
軸的直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449863280.png)
與橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449692313.png)
相交于不同的兩點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449894289.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449910333.png)
,過
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449894289.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449910333.png)
兩點作圓心為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449957399.png)
的圓,使橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449692313.png)
上的其余點均在圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449957399.png)
外.求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450035633.png)
的面積
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450035321.png)
的最大值.
(1)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450066745.png)
;(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450082386.png)
.
試題分析:(1)根據(jù)題干條件求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450082283.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450113249.png)
的值,進(jìn)而求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450128299.png)
的值,從而確定橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449692313.png)
的標(biāo)準(zhǔn)方程;(2)設(shè)點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449957399.png)
的坐標(biāo)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450160589.png)
,并設(shè)橢圓上任意一點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450191357.png)
的坐標(biāo)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450222533.png)
,求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450238598.png)
,根據(jù)題中條件得到點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450253656.png)
的坐標(biāo)使得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450238598.png)
取得最小值,從而得出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450300476.png)
,最后再求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450035633.png)
面積
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450035321.png)
的表達(dá)式,結(jié)合二次函數(shù)或基本不等式求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450035321.png)
的最大值.
試題解析:(1)設(shè)所求橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449692313.png)
的標(biāo)準(zhǔn)方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240434503941107.png)
,
由題意得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450425906.png)
,解的
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450440371.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450456454.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450487732.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450503195.png)
所求橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449692313.png)
的標(biāo)準(zhǔn)方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450066745.png)
;
(2)由橢圓的對稱性,可設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450550717.png)
,又設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450565659.png)
是橢圓上任意一點,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240434505812792.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450612531.png)
,
所以當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450628448.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450238598.png)
取最小值
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450659499.png)
,
又由題意得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449894289.png)
是橢圓上任意一點到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043449957399.png)
的距離最小的點,
設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450253656.png)
,因此當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450721357.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450238598.png)
取最小值,
又因
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450752556.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450300476.png)
,
由對稱性知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450784707.png)
,故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450784650.png)
,所以
S
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240434507993704.png)
,
所以當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450815508.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450035633.png)
的面積
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450035321.png)
取得最大值
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824043450082386.png)
.
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