如圖,斜率為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055180206.png)
的直線過拋物線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055227818.png)
的焦點(diǎn),與拋物線交于兩點(diǎn)A、B, M為拋物線弧AB上的動(dòng)點(diǎn).
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240300552895828.png)
(Ⅰ)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055305502.png)
,求拋物線的方程;
(Ⅱ)求△ABM面積
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055320301.png)
的最大值.
(I)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055336526.png)
;(II)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055383815.png)
.
試題分析:(I) 寫出直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055398396.png)
的方程
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055461819.png)
聯(lián)立
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055492986.png)
,消去
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055508310.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055523812.png)
.根據(jù)弦長公式
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055695849.png)
,解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055710433.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055336526.png)
.(II)根據(jù)(I) 設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055757927.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055773400.png)
到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055398396.png)
的距離:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240300558041163.png)
而M在直線AB上方,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055866942.png)
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240300558821193.png)
則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055944916.png)
,所以當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055960487.png)
時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055991321.png)
取最大值
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030056007551.png)
此時(shí)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055383815.png)
.
試題解析:(I) 根據(jù)條件得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055461819.png)
則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055492986.png)
,消去
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055508310.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055523812.png)
.
令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030056100858.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030056116589.png)
,又拋物線定義得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055695849.png)
根據(jù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030056147507.png)
,解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055710433.png)
,拋物線方程
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055336526.png)
.
(II)由(I) 知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030056178619.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030056225815.png)
設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055757927.png)
則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055773400.png)
到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055398396.png)
的距離:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240300558041163.png)
由M在直線AB上方,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055866942.png)
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240300558821193.png)
,由(I)知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055944916.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030056334195.png)
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055960487.png)
時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055991321.png)
取最大值
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030056007551.png)
此時(shí)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030055383815.png)
.
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025818493382.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025818743297.png)
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的
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兩點(diǎn).點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025730118544.png)
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來源:不詳
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設(shè)橢圓
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:
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的左、右焦點(diǎn)分別是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025210879333.png)
、
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的中點(diǎn)為
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(
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為坐標(biāo)原點(diǎn)),如圖.若拋物線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025210957372.png)
:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025210972507.png)
與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025210988331.png)
軸的交點(diǎn)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025210941304.png)
,且經(jīng)過
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025210879333.png)
、
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兩點(diǎn).
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(Ⅰ)求橢圓
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的方程;
(Ⅱ)設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025211066725.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025211082357.png)
為拋物線
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上的一動(dòng)點(diǎn),過點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025211082357.png)
作拋物線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025210957372.png)
的切線交橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025210848339.png)
于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025211253290.png)
、
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面積的最大值.
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來源:不詳
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知橢圓
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的左右焦點(diǎn)為F
1,F(xiàn)
2,離心率為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021737791414.png)
,以線段F
1 F
2為直徑的圓的面積為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021737807309.png)
, (1)求橢圓的方程;(2) 設(shè)直線l過橢圓的右焦點(diǎn)F
2(l不垂直坐標(biāo)軸),且與橢圓交于A、B兩點(diǎn),線段AB的垂直平分線交x軸于點(diǎn)M(m,0),試求m的取值范圍.
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來源:不詳
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已知橢圓
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與雙曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030351273372.png)
有共同的焦點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030351289587.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030351304579.png)
,橢圓的一個(gè)短軸端點(diǎn)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030351351309.png)
,直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030351367422.png)
與雙曲線的一條漸近線平行,橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030351258336.png)
與雙曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030351273372.png)
的離心率分別為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030351414406.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030351414412.png)
取值范圍為( )
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