已知圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240233222481179.png)
,圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240233222641118.png)
,動圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322279289.png)
與已知兩圓都外切.
(1)求動圓的圓心
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322279289.png)
的軌跡
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322310318.png)
的方程;
(2)直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322388594.png)
與點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322279289.png)
的軌跡
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322310318.png)
交于不同的兩點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322466300.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322482309.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322482396.png)
的中垂線與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322498310.png)
軸交于點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322513357.png)
,求點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322513357.png)
的縱坐標(biāo)的取值范圍.
試題分析:(1)兩圓外切,則兩圓圓心之間的距離等于兩圓的半徑之和,由此得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240233226691198.png)
將兩式相減得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240233226851072.png)
由雙曲線的定義可得軌跡
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322310318.png)
的方程.
(2)將直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322716280.png)
的方程
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322732528.png)
代入軌跡
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322310318.png)
的方程,利用根與系數(shù)的關(guān)系得到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322466300.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322482309.png)
的中點(diǎn)的坐標(biāo)(用
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322794312.png)
表示),從而得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322482396.png)
的中垂線的方程。再令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322825367.png)
得點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322513357.png)
的縱坐標(biāo)(用
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322794312.png)
表示).根據(jù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322794312.png)
的范圍求出點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322513357.png)
的縱坐標(biāo)的取值范圍.
本小題中要利用
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322919344.png)
及與雙曲線右支相交求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322794312.png)
的范圍,這是一個易錯之處.
試題解析:(1)已知兩圓的圓心、半徑分別為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240233229501606.png)
設(shè)動圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322279289.png)
的半徑為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322981260.png)
,由題意知:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240233226691198.png)
則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240233226851072.png)
所以點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322279289.png)
在以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023323044449.png)
為焦點(diǎn)的雙曲線的右支上,其中
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023323059701.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023323075395.png)
由此得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322310318.png)
的方程為:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322622796.png)
4分
(2)將直線代入雙曲線方程并整理得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023323122829.png)
設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023323153961.png)
的中點(diǎn)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023323168723.png)
依題意,直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322716280.png)
與雙曲線右支交于不同兩點(diǎn),故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240233232003021.png)
且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240233232001099.png)
則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322482396.png)
的中垂線方程為:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240233232311026.png)
令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023322825367.png)
得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240233232621295.png)
12分
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的方程;
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與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022544372266.png)
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已知
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為拋物線
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的焦點(diǎn),拋物線上點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021251453691.png)
滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021251468533.png)
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(Ⅰ)求拋物線
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的方程;
(Ⅱ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021251546399.png)
點(diǎn)的坐標(biāo)為(
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021251562248.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021251577262.png)
),過點(diǎn)F作斜率為
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的直線與拋物線交于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021251593300.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021251609309.png)
兩點(diǎn),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021251593300.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021251609309.png)
兩點(diǎn)的橫坐標(biāo)均不為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021251655256.png)
,連結(jié)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021251671478.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021251687473.png)
并延長交拋物線于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021251718313.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021251718315.png)
兩點(diǎn),設(shè)直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021251733405.png)
的斜率為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021251749370.png)
,問
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021251765462.png)
是否為定值,若是求出該定值,若不是說明理由.
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來源:不詳
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過橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024613841636.png)
的左焦點(diǎn)作互相垂直的兩條直線,分別交橢圓于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024613841584.png)
四點(diǎn),則四邊形
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024613857526.png)
面積的最大值與最小值之差為( )
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科目:高中數(shù)學(xué)
來源:不詳
題型:單選題
已知橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240201105431163.png)
的離心率為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020110559449.png)
,雙曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020110575679.png)
的漸近線與橢圓有四個交點(diǎn),以這四個交點(diǎn)為頂點(diǎn)的四邊形的面積為16,則橢圓的方程為( )
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