题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164201-55903.jpg)
(Ⅰ)求证:FM1⊥FN1:
(Ⅱ)记△FMM1、、△FM1N1、△FN N1的面积分别为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164201-52171.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164202-25457.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164202-55808.jpg)
答案
解析
(1) 证法1:由抛物线的定义得
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164202-52988.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164202-25457.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164202-55808.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164202-44399.gif)
如图,设准线l与x的交点为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164203-32331.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164203-52158.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164203-92947.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164204-75975.gif)
而
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164204-55354.gif)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164204-70155.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164204-90016.gif)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164205-98537.gif)
证法2:依题意,焦点为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164205-36467.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164205-15910.gif)
设点M,N的坐标分别为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164206-81685.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164206-64308.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164206-73816.gif)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164206-80854.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164207-45553.gif)
于是,
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164207-78341.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164207-42024.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164207-90189.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164205-98537.gif)
(Ⅱ)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164208-23715.gif)
证法1:设
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164208-28915.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164208-18784.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164209-73916.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164209-49880.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164209-75649.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164209-85438.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164210-95784.gif)
将
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164210-27820.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164210-10895.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164202-25457.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164202-55808.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164211-51593.gif)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164208-23715.gif)
证法2:如图,设直线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164211-70809.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164211-45295.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164211-73924.gif)
则由抛物线的定义得
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164212-22967.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164212-38242.gif)
于是
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164212-49900.gif)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164212-68322.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164213-98397.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164213-39916.gif)
由(I)的结论,得
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164213-66860.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164213-34865.gif)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164208-23715.gif)
核心考点
试题【如图,过抛物线y2=2PX(P>0)的焦点F的直线与抛物线相交于M、N两点,自M、N向准线L作垂线,垂足分别为M1、N1 (Ⅰ)求证:FM1⊥FN1】;主要考察你对抛物线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/20191025/20191025164148-40012.gif)
A.0.5 | B.1 | C.2 | D.4 |
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164125-31938.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164126-78796.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164126-64212.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164120-59346.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164120-51702.gif)
A.-3 | B.-4 | C.5 | D.6 |
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164104-18373.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164104-10573.gif)
A
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164104-24411.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164031-76932.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164031-57937.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025164031-48409.gif)
A 1 B 2 C 3 D 4
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