题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154910-40560.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154910-21512.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154911-44533.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154911-50057.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154911-93135.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154911-44533.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154911-85574.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154912-21341.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154912-33059.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154912-47141.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154912-14271.png)
(I)求椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154911-44533.png)
(Ⅱ)求线段
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154912-67759.png)
(Ⅲ)当线段
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154912-67759.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154911-44533.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154913-84957.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154913-63453.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154913-11351.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154913-84957.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154913-12628.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154913-61546.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154912-67759.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154914-83369.png)
(Ⅲ)当线段MN的长度最小时,在椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154911-44533.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154913-84957.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154914-47202.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154913-11351.png)
解析
试题分析:(1)由已知得,椭圆C的左顶点为A(-2,0),上顶点为D(0,1,由此能求出椭圆C的方程.(2)设直线AS的方程为y=k(x+2),从而M(
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154914-47047.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154914-13930.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154914-47047.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154915-62724.png)
(3)在第二问的基础上确定了直线BS的斜率得到直线方程,利用点到直线的距离得到l‘,然后得到分析方程组的解的个数即为满足题意的点的个数。
解:(I)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154915-98167.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154915-12008.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154911-44533.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154913-12628.png)
(Ⅱ)直线AS的斜率
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154915-77856.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154915-86302.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154916-47131.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154916-91500.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154916-50015.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154916-75570.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154916-53918.png)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154917-41317.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154917-16131.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154917-65430.png)
从而
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154917-75850.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154918-56188.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154918-58566.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154918-61425.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154918-67530.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154918-81625.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154919-98567.png)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154919-71801.png)
当且仅当
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154919-49384.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154919-83525.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154913-61546.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154912-67759.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154914-83369.png)
(Ⅲ)由(Ⅱ)可知,当
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154912-67759.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154919-83525.png)
此时
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154920-35255.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154920-50736.png)
要使椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154911-44533.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154913-84957.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154914-47202.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154913-11351.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154913-84957.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154920-35255.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154920-86154.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154913-84957.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154920-35255.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154920-35255.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154920-86154.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154920-86801.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154920-34106.png)
则由
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154921-33356.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154921-33849.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154921-38036.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154921-33849.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154921-86080.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154922-32016.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154922-40045.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154921-38036.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154922-63937.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154922-18318.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154922-77176.png)
综上:当线段MN的长度最小时,在椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154911-44533.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154913-84957.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154914-47202.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154913-11351.png)
点评:解决该试题的关键是能利用椭圆的几何性质表述出|MN|,同时结合均值不等式求解最小值。
核心考点
试题【(本小题14分)已知直线经过椭圆的左顶点A和上顶点D,椭圆的右顶点为,点是椭圆上位于轴上方的动点,直线与直线分别交于两点。(I)求椭圆的方程;(Ⅱ)求线段的长度】;主要考察你对椭圆的定义与方程等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154904-99598.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154904-53548.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154904-44014.png)
A.两个交点 | B.一个交点 | C.没有交点 | D.无法判断 |
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154856-51713.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154856-65125.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154856-76444.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154856-13003.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154856-36518.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154856-52908.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154856-65125.png)
(Ⅰ)求椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154856-65125.png)
(Ⅱ)已知直线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154857-97967.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154856-65125.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154857-97967.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154856-65125.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154849-43055.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154849-46406.png)
(Ⅰ)求此双曲线的渐近线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154849-80595.png)
(Ⅱ)若
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154849-10603.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154849-80668.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154849-80595.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154850-41680.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154850-13617.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154850-30682.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154840-89532.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154840-69371.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154840-54964.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154840-17533.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154840-83622.png)
(Ⅰ)求椭圆M的标准方程;
(Ⅱ) 设直线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154840-98676.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154841-52133.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154841-66913.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154841-74959.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154830-24366.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154830-93448.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154830-90593.png)
A.![]() | B.-1 | C.2 | D.7 |
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