题目
题型:不详难度:来源:
已知抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140215-26486.gif)
(1)若直线l与抛物线交于两点A、B,且
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140215-89392.gif)
(2)若C、D两点在抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140215-26486.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140215-61957.gif)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140215-39608.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140216-89407.gif)
直线CD的方程可化为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140216-47405.gif)
解析
解 (1) 设动点M的坐标为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140216-85525.gif)
∵抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140217-88263.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140217-80896.gif)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140217-84062.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140217-76417.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140218-41093.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140215-39608.gif)
又当M与原点重合时,直线l与x轴重合,故
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140216-89407.gif)
∴所求动点M的轨迹方程是
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140215-39608.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140216-89407.gif)
(2) 设点C、D的坐标为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140218-99247.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140218-64408.gif)
∵C、D在抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140217-88263.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140218-66457.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140219-36002.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140219-23763.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140219-28039.gif)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140219-30651.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140220-17298.gif)
∵点C、D的坐标为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140218-99247.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140218-64408.gif)
∴直线CD的一个法向量是
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140220-29631.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140220-45318.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140221-22429.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140221-99365.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140221-40220.gif)
又抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140217-88263.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140221-64989.gif)
∴直线CD的方程可化为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140216-47405.gif)
∴直线CD恒过定点,且定点坐标为(2,0). ………………………12分
核心考点
试题【(本题满分12分)本题共有2个小题,第1小题满分5分,第2小题满分7分.已知抛物线,F是焦点,直线l是经过点F的任意直线.(1)若直线l与抛物线交于两点A、B,】;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/20191025/20191025140209-59453.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140209-82894.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140210-73881.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140210-88357.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140210-57626.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140200-81196.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140200-72314.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140201-68659.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140201-44671.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140201-82237.gif)
( )
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140155-85958.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140155-71845.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140156-92875.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140156-35473.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140156-91473.png)
设抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140109-63029.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140110-95835.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140110-96177.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140110-96177.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140110-46854.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140110-27820.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140110-24751.png)
(1)求抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140111-75430.png)
(2)设
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140111-94253.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140111-80448.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140111-97724.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140111-75430.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140111-35842.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140111-79975.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140112-84116.png)
(3)①对给定的定点
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140112-24897.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140112-91859.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140112-24897.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140111-75430.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140111-35842.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140110-46854.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140112-96544.png)
②对
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140112-93974.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140112-24897.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140111-75430.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140111-35842.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140110-46854.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140112-96544.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140105-16362.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140105-69531.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140105-91780.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140106-42177.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140106-91977.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140106-39335.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025140106-17504.gif)
A.![]() | B.![]() | C.![]() | D.![]() |
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