已知橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236391361183.png)
過(guò)點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639152552.png)
,離心率為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639167453.png)
.
(Ⅰ)求橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639167313.png)
的方程;
(Ⅱ)過(guò)點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639183494.png)
且斜率為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639198312.png)
(
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639214418.png)
)的直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639230280.png)
與橢圓
![](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)
的中點(diǎn)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639417289.png)
.記直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639432365.png)
的斜率為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639448350.png)
,求證:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639464396.png)
為定值.
(Ⅰ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639495636.png)
;(Ⅱ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639510550.png)
試題分析:(Ⅰ)根據(jù)條件可得以下方程組:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236395261200.png)
,解這個(gè)方程組求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639542283.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639573299.png)
的值便得橢圓的方程;(Ⅱ)將
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639448350.png)
用
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639198312.png)
表示出來(lái),這樣
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639464396.png)
就是一個(gè)只含
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639198312.png)
的式子,將該式化簡(jiǎn)即可.那么如何用
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639198312.png)
來(lái)表示
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639448350.png)
?
設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639713632.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639729637.png)
.因?yàn)锳(2,0),所以直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639744523.png)
的方程分別為:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236397601138.png)
.
令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639339382.png)
得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236397911253.png)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639401513.png)
的中點(diǎn)為:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236398221136.png)
由此得直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639432365.png)
的斜率為:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236398542295.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236398692966.png)
①
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236398859112.png)
再設(shè)直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639230280.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639916610.png)
,代入橢圓方程
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639495636.png)
得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236399321006.png)
設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639713632.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639729637.png)
,則由韋達(dá)定理得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236399941165.png)
代入①式,便可將
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639448350.png)
用
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639198312.png)
表示出來(lái),從而得到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639464396.png)
的值.
試題解析:(Ⅰ)由題設(shè):
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236395261200.png)
,解之得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023640244531.png)
,所以橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639167313.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639495636.png)
4分
(Ⅱ)設(shè)直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639230280.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639916610.png)
代入橢圓方程
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639495636.png)
得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236399321006.png)
設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639713632.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639729637.png)
,則由韋達(dá)定理得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236399941165.png)
直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639744523.png)
的方程分別為:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236397601138.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236404469160.png)
令,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023639339382.png)
得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236397911253.png)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236405091138.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236405242960.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236405403065.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240236406342316.png)
13分
練習(xí)冊(cè)系列答案
相關(guān)習(xí)題
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
已知橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030508493313.png)
的中心在坐標(biāo)原點(diǎn),焦點(diǎn)在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030508509266.png)
軸上,橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030508493313.png)
上的點(diǎn)到焦點(diǎn)距離的最大值為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030508525287.png)
,最小值為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030508556206.png)
.
(Ⅰ)求橢圓方程;
(Ⅱ)若直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030508603886.png)
與橢圓交于不同的兩點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030508618399.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030508634357.png)
,且線段
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030508649513.png)
的垂直平分線過(guò)定點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030508649643.png)
,求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030508665312.png)
的取值范圍.
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
已知圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025540465807.png)
直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025540481821.png)
與圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025540496336.png)
相切,且交橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240255405121188.png)
于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025540528460.png)
兩點(diǎn),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025540543249.png)
是橢圓的半焦距,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025540574488.png)
,
(Ⅰ)求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025540652337.png)
的值;
(Ⅱ)O為坐標(biāo)原點(diǎn),若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025540699610.png)
求橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025540715372.png)
的方程;
(Ⅲ) 在(Ⅱ)的條件下,設(shè)橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025540715372.png)
的左右頂點(diǎn)分別為A,B,動(dòng)點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025540762963.png)
,直線AS,BS與直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025540777565.png)
分別交于M,N兩點(diǎn),求線段MN的長(zhǎng)度的最小值.
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
設(shè)拋物線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024617881911.png)
的焦點(diǎn)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024617913302.png)
,準(zhǔn)線為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024617913280.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024617944531.png)
,以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024617959399.png)
為圓心的圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024617959399.png)
與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024617913280.png)
相切于點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024617991333.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024617991333.png)
的縱坐標(biāo)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024618022441.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024618037559.png)
是圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024617959399.png)
與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024618069266.png)
軸除
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024617913302.png)
外的另一個(gè)交點(diǎn).
(I)求拋物線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024618100313.png)
與圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024617959399.png)
的方程;
( II)已知直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024618131872.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024618147297.png)
與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024618100313.png)
交于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024618209423.png)
兩點(diǎn),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024618147297.png)
與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024617913280.png)
交于點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024618349315.png)
,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024618365573.png)
, 求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024618365557.png)
的面積.
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
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.
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
經(jīng)過(guò)點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552125503.png)
且與直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552140370.png)
相切的動(dòng)圓的圓心軌跡為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552156399.png)
.點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552187413.png)
在軌跡
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552156399.png)
上,且關(guān)于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552218310.png)
軸對(duì)稱,過(guò)線段
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552234385.png)
(兩端點(diǎn)除外)上的任意一點(diǎn)作直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552250280.png)
,使直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552250280.png)
與軌跡
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552156399.png)
在點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552281315.png)
處的切線平行,設(shè)直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552250280.png)
與軌跡
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552156399.png)
交于點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552374420.png)
.
(1)求軌跡
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552156399.png)
的方程;
(2)證明:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552390665.png)
;
(3)若點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552281315.png)
到直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552437396.png)
的距離等于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552452629.png)
,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552452544.png)
的面積為20,求直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022552484398.png)
的方程.
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
已知點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022434187616.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022434203644.png)
是拋物線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022434203525.png)
上相異兩點(diǎn),且滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022434249508.png)
.
(Ⅰ)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022434265396.png)
的中垂線經(jīng)過(guò)點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022434296533.png)
,求直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022434265396.png)
的方程;
(Ⅱ)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022434265396.png)
的中垂線交
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022434343262.png)
軸于點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022434374399.png)
,求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022434390601.png)
的面積的最大值及此時(shí)直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022434265396.png)
的方程.
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
設(shè)橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240213359131084.png)
的左右頂點(diǎn)分別為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021335928742.png)
,離心率
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021335944551.png)
.過(guò)該橢圓上任一點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021335959289.png)
作
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021335975486.png)
軸,垂足為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021335991333.png)
,點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021336069313.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021336084399.png)
的延長(zhǎng)線上,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021336100588.png)
.
(1)求橢圓的方程;
(2)求動(dòng)點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021336069313.png)
的軌跡
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021336209318.png)
的方程;
(3)設(shè)直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021336225401.png)
(
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021336069313.png)
點(diǎn)不同于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021336256423.png)
)與直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021336256383.png)
交于點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021336287303.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021336303315.png)
為線段
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021336303360.png)
的中點(diǎn),試判斷直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021336318405.png)
與曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021336209318.png)
的位置關(guān)系,并證明你的結(jié)論.
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:填空題
設(shè)拋物線的頂點(diǎn)在原點(diǎn),準(zhǔn)線方程為x =﹣2,則拋物線的方程是 .
查看答案和解析>>