設(shè)雙曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157673312.png)
以橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157688792.png)
的兩個(gè)焦點(diǎn)為焦點(diǎn),且雙曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157673312.png)
的一條漸近線是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157720509.png)
,
(1)求雙曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157673312.png)
的方程;
(2)若直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157735926.png)
與雙曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157673312.png)
交于不同兩點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157766431.png)
,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157766431.png)
都在以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157798534.png)
為圓心的圓上,求實(shí)數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157798316.png)
的取值范圍.
(1)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157813715.png)
;(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240241578291005.png)
試題分析:(1)雙曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157673312.png)
和橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157688792.png)
共焦點(diǎn),故可設(shè)其方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240241578761114.png)
,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157876575.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157891562.png)
,聯(lián)立解
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157907396.png)
;(2)直線和圓錐曲線的位置關(guān)系問題,一般根據(jù)已知條件結(jié)合韋達(dá)定理列方程來確定參數(shù)的值或取值范圍,因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157766431.png" style="vertical-align:middle;" />在以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157798534.png)
為圓心的圓上,根據(jù)垂徑定理,連接圓心和弦
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157954386.png)
的中點(diǎn)的直線必垂直于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157954386.png)
,∴將直線和雙曲線聯(lián)立,得關(guān)于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157985266.png)
的一元二次方程且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158000426.png)
,得關(guān)于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158016446.png)
的不等式,利用韋達(dá)定理確定弦
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157954386.png)
的中點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158047316.png)
坐標(biāo),利用
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158063624.png)
列式,得關(guān)于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158016446.png)
的方程,與不等式聯(lián)立消去
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158094312.png)
,得關(guān)于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158094337.png)
的不等式,解之可得.
試題解析:(1)依題雙曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158110312.png)
的兩個(gè)焦點(diǎn)分別為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158125465.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158141464.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158156361.png)
,又雙曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158110312.png)
的一條漸近線是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157720509.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158188592.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158203614.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158219196.png)
雙曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158110312.png)
的方程為:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024157813715.png)
;
(2)設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158250634.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158266638.png)
,
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240241582811152.png)
,消去
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158297308.png)
整理得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240241583121088.png)
,依題意得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240241583281646.png)
(*),設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158328389.png)
的中點(diǎn)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158359649.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158375991.png)
,
又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158390231.png)
點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158390314.png)
在直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158406601.png)
上,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240241584221023.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158437998.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158390231.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158468424.png)
兩點(diǎn)都在以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158484527.png)
為圓心的同一圓上,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158500592.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158515523.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240241585311089.png)
,整理得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158546757.png)
,代人(*)式得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240241585622373.png)
解得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158578460.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158578620.png)
,
又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158593859.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158609581.png)
,故所求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024158624339.png)
的取值范圍是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240241578291005.png)
.
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來源:不詳
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已知拋物線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030523797545.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030523875582.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030523891309.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030524093328.png)
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如圖示:已知拋物線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025621290624.png)
的焦點(diǎn)為
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的切線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025621446314.png)
、
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與
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.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/2014082402562152411100.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025621540297.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025621556352.png)
時(shí),求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025621649419.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025621665640.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236391361183.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639152552.png)
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(
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639230280.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639167313.png)
相交于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639292421.png)
兩點(diǎn),直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639308410.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639323395.png)
分別交直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639339382.png)
于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639354399.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639386357.png)
兩點(diǎn),線段
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639401513.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639417289.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639432365.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639448350.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639464396.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024611704280.png)
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交
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024611751423.png)
兩點(diǎn),若點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024611766520.png)
是
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過拋物線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022539504525.png)
的焦點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022539520302.png)
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的直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022539567280.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022539582415.png)
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過直線
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上一點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020258184289.png)
作圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020258215919.png)
的切線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020258230421.png)
,若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020258230421.png)
關(guān)于直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020258262280.png)
對稱,則點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020258184289.png)
到圓心
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020258293313.png)
的距離為
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