已知椭圆
的左右顶点分别为
,左右焦点为分别为
,焦距为2,离心率为
.
(Ⅰ)求椭圆
的标准方程;
(Ⅱ)若
为椭圆上一动点,直线
过点
且与
轴垂直,
为直线
与
的交点,
为直线
与直线
的交点,求证:点
在一个定圆上.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/851a5d6ec23256f9b4a9e98aa92945fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00442d96d695db2c58bf1fb7165fca94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
(Ⅰ)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(Ⅱ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edc48974114e23f5a801843710c7ae21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d0ff310aabd2282b539537ebed3f788.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/183b6a0cef4256c9696a5bca31053da5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
更新时间:2018-04-02 09:26:41
|
【知识点】 椭圆中的定点、定值
相似题推荐
解答题-问答题
|
适中
(0.65)
解题方法
【推荐1】折纸又称“工艺折纸”,是一种把纸张折成各种不同形状物品的艺术活动,在我国源远流长. 某些折纸活动蕴含丰富的数学内容,例如:用圆形纸片,按如下步骤折纸,可折出一个椭圆.
![](https://img.xkw.com/dksih/QBM/2021/1/28/2646018424455168/2648237748101120/STEM/8b52d5d840f142899dbde5e76bca0822.png?resizew=263)
![](https://img.xkw.com/dksih/QBM/2021/1/28/2646018424455168/2648237748101120/STEM/2b4845d0-1061-4c4f-81ac-6a9c09a074c9.png?resizew=323)
步骤1:设圆心是F,在圆内不是圆心处取一点,标记为E;
步骤2:把纸片对折,使圆周正好通过点E,此时圆周上与点E重合的点标记为G;
步骤3:把纸片展开,于是就留下一条折痕,此时GF与折痕交于点P;
步骤4:不断重复步骤2和3,能得到越来越多条的折痕和越来越多的交点P,所有交点P组成的图形便是一个椭圆.
现已知圆形纸片的半径为4,定点E到圆心F的距离为2,
为EF中点,所有交点P组成的椭圆记为
.
(1)以EF所在的直线为x 轴,以O为原点建立平面直角坐标系,求椭圆
的标准方程;
(2)设直线
与椭圆
交于A,
两点,且
,试问点
到直线
的距离是否为定值?如果是定值,则求该定值;如果不是定值,则说明理由.
![](https://img.xkw.com/dksih/QBM/2021/1/28/2646018424455168/2648237748101120/STEM/8b52d5d840f142899dbde5e76bca0822.png?resizew=263)
![](https://img.xkw.com/dksih/QBM/2021/1/28/2646018424455168/2648237748101120/STEM/2b4845d0-1061-4c4f-81ac-6a9c09a074c9.png?resizew=323)
步骤1:设圆心是F,在圆内不是圆心处取一点,标记为E;
步骤2:把纸片对折,使圆周正好通过点E,此时圆周上与点E重合的点标记为G;
步骤3:把纸片展开,于是就留下一条折痕,此时GF与折痕交于点P;
步骤4:不断重复步骤2和3,能得到越来越多条的折痕和越来越多的交点P,所有交点P组成的图形便是一个椭圆.
现已知圆形纸片的半径为4,定点E到圆心F的距离为2,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(1)以EF所在的直线为x 轴,以O为原点建立平面直角坐标系,求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3825ccc273ef9a672a606432d165b866.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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解答题-问答题
|
适中
(0.65)
名校
解题方法
【推荐2】已知椭圆
过点
,且离心率
为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8eff998d034284391ca064755fa6bf1b.png)
(1)求椭圆
的方程;
(2)
、
是椭圆上的两个动点,如果直线
的斜率与
的斜率互为相反数,证明直线
的斜率为定值,并求出这个定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dae13b2a96f91ab64fb4948de2b0ae10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24261b8fc52ec9a02831a4742c0ba6fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8eff998d034284391ca064755fa6bf1b.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9f79d7939c88e9702962e5917cad290.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79ff575e55857af133edb24c8e61504f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0215e13a9fb5574d5194aeb9507a98aa.png)
您最近一年使用:0次