题目
题型:不详难度:来源:
(1) 求抛物线C的方程;
(2)在抛物线C上是否存在点P, 使得过点P
的直线交C于另一点Q,满足PF⊥QF, 且
PQ与C在点P处的切线垂直.若存在,求出
点P的坐标; 若不存在,请说明理由.
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023945-16246.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023945-33177.png)
故所求抛物线C的方程为x2 = 4y .
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023945-12431.png)
(2) 解:设P(
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023945-93724.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023945-23927.png)
直线PQ的方程是:
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023946-12337.png)
将上式代入抛物线C的方程, 得:
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023946-96113.png)
故 x1+x2=
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023946-44135.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023946-44135.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023946-90676.png)
而
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023947-93636.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023947-82877.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023947-93636.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023947-82877.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023946-90676.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023946-90676.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023947-50147.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023946-90676.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023947-50147.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023947-73722.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023947-59051.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023948-10811.png)
故 y1=4, 此时, 点P的坐标是(±4,4) . 经检验, 符合题意.
所以, 满足条件的点P存在, 其坐标为P(±4,4). ………………(15分)
解析
核心考点
试题【(本小题满分15分) 已知抛物线C的顶点在原点, 焦点为F(0,1).(1) 求抛物线C的方程;(2)在抛物线C上是否存在点P, 使得过点P的直线交C于另一点Q】;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/20191026/20191026023940-82065.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023934-65211.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023935-74363.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023935-21252.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023935-67707.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023934-65211.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023914-80647.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023914-95967.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023914-33484.png)
(Ⅰ)求
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023914-12444.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023915-50915.png)
(Ⅱ)若过点
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023915-59646.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023915-73930.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023915-83432.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023916-35591.jpg)
(1)当m为何值时,曲线C表示圆;
(2)若曲线C与直线x+2y-4=0交于M、N两点,且OM⊥ON(O为坐标原点),求m的值。
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023900-77623.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023900-61597.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023900-61597.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023900-91565.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023900-79710.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023900-66504.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023900-61597.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023900-79710.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023900-59564.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023900-66606.png)
(Ⅰ)求曲线
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023900-66606.png)
(Ⅱ)过点
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023900-91720.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023901-91236.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023901-80793.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023900-66606.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023901-80522.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023901-75362.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023901-94502.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023901-42659.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026023901-82929.png)
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