题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171729-56704.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171729-45068.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171729-82180.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171729-15885.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171729-87161.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171730-67719.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171730-95847.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171730-16958.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171731-29953.png)
A.充分非必要条件 | B.充要条件 |
C.必要非充分条件 | D.非充分非必要条件 |
答案
解析
解:由
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171731-55529.png)
设点A,B的坐标分别为(x1,y1),(x2,y2)
(I)当直线l有存在斜率时,设直线方程为y=kx+b,显然k≠0且b≠0.
联立方程得:
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171731-66558.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171731-57780.png)
又由OA⊥OB得x1x2+y1y2=0,
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171731-90672.png)
故直线l的方程为:y=kx-2pk=k(x-2p),故直线过定点(2p,0)
(II)当直线l不存在斜率时,设它的方程为x=m,显然m>0
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171731-96486.png)
又由OA⊥OB得x1x2+y1y2=0,即m2-2m=0,解得m=0(舍去)或m=2
可知直线l方程为:x=2,故直线过定点(2,0)
综合(1)(2)可知,满足条件的直线过定点(2,0).
由“直线AB恒过定点(2p,0)”推
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171731-55529.png)
设l:x=ty+2p代入抛物线y2=2px消去x得,
y2-2pty-4p2=0,设A(x1,y1),B(x2,y2)
则y1+y2=2pt,y1y2=-4p2
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171731-55529.png)
=t2y1y2+2pt(y1+y2)+4p2+y1y2
=-4p2t2+4p2t2+4p2-4p2=0.
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171731-55529.png)
故选B.
核心考点
试题【已知、是抛物线(>0)上异于原点的两点,则“=0”是“直线恒过定点()”的( )A.充分非必要条件B.充要条件C.必要非充分条件D.非充分非必要条件】;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/20191025/20191025171725-50594.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171725-50013.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171725-48167.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171725-54543.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171726-38568.png)
A.![]() | B.8 | C.16 | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171711-82048.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171712-46657.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171712-46657.png)
A.![]() | B.![]() | C.![]() | D.0 |
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171639-83743.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171639-84092.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171639-83743.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171639-74148.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171639-52827.png)
A.![]() | B.![]() | C.2 | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171635-11840.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171635-33269.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171635-35001.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171635-69886.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171636-66145.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171636-71495.gif)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171624-65550.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171624-43675.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171624-65136.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171624-77047.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171625-25450.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171625-71068.gif)
(1)当点
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171624-77047.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171625-25450.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171625-23686.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171625-30944.gif)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171626-96165.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171624-65136.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171626-30217.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171626-99383.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025171626-30217.gif)
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