题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235426-13925.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235426-91564.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235426-89329.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235426-82927.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235426-30553.png)
(Ⅰ)求椭圆C的方程;
(Ⅱ)设n 为过原点的直线,
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235426-91835.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235427-18619.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235426-91835.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235427-80287.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235426-91835.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235427-83401.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235427-89004.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235426-91835.png)
解析
试题分析:(Ⅰ)由
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235427-18822.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235427-11701.png)
又b2=a2-c2 ③
由 ①,②,③解得a2=4,b2=3,
故椭圆C的方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235427-83401.png)
(Ⅱ) 设A,B两点的坐标分别为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235427-78730.png)
假设使
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235427-89004.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235428-79268.png)
(i) 当l不垂直于x轴时,设l的方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235428-70801.png)
由l与n垂直相交于P点且
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235427-18619.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235428-36894.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235428-87080.png)
将y=kx+m代入椭圆方程,得(3+4k2)x2+8kmx+(4m2-12)=0,
由求根公式可得x1+x2=
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235428-93339.png)
x1+x2=
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235428-98960.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235428-44027.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235429-81263.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235429-67468.png)
将④,⑤代入上式并化简得
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235429-29931.png)
将
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235429-18585.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235429-16483.png)
即此时直线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235426-91835.png)
(ii)当
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235426-91835.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235429-44850.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235429-54625.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235426-91835.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235430-77312.png)
则A,B两点的坐标为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235430-90876.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235430-36220.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235430-48631.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235430-94802.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235431-27784.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235431-37960.png)
∴ 此时直线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235426-91835.png)
综上可知,使
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235427-89004.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235426-91835.png)
点评:椭圆的概念和性质,仍将是今后命题的热点,定值、最值、范围问题将有所加强;利用直线、弦长、圆锥曲线三者的关系组成的各类试题是解析几何中长盛不衰的主题,其中求解与相交弦有关的综合题仍是今后命题的重点;与其它知识的交汇(如向量、不等式)命题将是今后高考命题的一个新的重点、热点.
核心考点
试题【如图,椭圆的顶点为,焦点为,. (Ⅰ)求椭圆C的方程;(Ⅱ)设n 为过原点的直线,是与n垂直相交于P点,与椭圆相交于A, B两点的直线,.是否存在上述直线使成立】;主要考察你对曲线与方程的应用等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235416-68315.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235417-77680.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235417-80175.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235417-13391.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235417-47953.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235417-21675.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235417-26556.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235418-92370.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235418-56625.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235418-60683.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235418-42399.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235419-64241.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235357-61045.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235357-29240.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235357-69221.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235357-71437.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235358-63015.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235358-36763.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235358-93054.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235359-76672.png)
(I)求椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235358-93054.png)
(II)当
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235359-52165.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235346-29491.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235346-25132.png)
(Ⅰ)求双曲线的方程;
(Ⅱ)若过点
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235347-13771.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235347-56443.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235347-63377.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235347-56524.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235347-56524.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235347-47567.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235340-14481.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235340-57970.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235340-69661.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235322-91506.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235322-56904.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235323-73752.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235323-45606.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235323-84038.png)
(I)求椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235322-91506.png)
(Ⅱ)设椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235322-91506.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235323-85395.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235324-32399.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235324-65117.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235324-26653.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235322-91506.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235325-93268.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235325-47566.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235325-61305.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235325-83542.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235325-61305.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235326-46820.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235326-69323.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235326-89594.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025235327-40195.png)
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