题目
题型:不详难度:来源:
已知定点
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103326-14005.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103326-58791.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103327-43383.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103327-29072.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103327-90355.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103327-18631.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103328-23404.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103328-37581.jpg)
(I)求动点
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103328-23404.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103328-60340.png)
(Ⅱ)设倾斜角为
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103328-20265.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103329-87039.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103327-90355.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103328-60340.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103329-55981.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103327-18631.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103329-14322.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103329-81200.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103330-34787.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103330-28495.png)
(Ⅱ) |PR|·|QR|的最小值为16
解析
(1)连CA,过C作CD⊥l1,垂足为D,由已知可得|CA|=|CD|,
∴点C的轨迹是以A为焦点,l1为准线的抛物线,
(2)设直线l2的方程为y=kx+1,
把直线方程与抛物线方程联立消去y得 x2-4kx-4=0.
结合韦达定理来表示关系式,以向量的数量积来表示模长的积,得到结论。
解法一:(Ⅰ)连CA,过C作CD⊥l1,垂足为D,由已知可得|CA|=|CD|,
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103330-77984.jpg)
∴点C的轨迹是以A为焦点,l1为准线的抛物线,
∴轨迹E的方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103330-28495.png)
(Ⅱ)设直线l2的方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103330-24466.png)
记P(x1,y1),Q(x2,y2),则
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103331-46554.png)
因为直线PA的斜率k≠O,易得点R的坐标为
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103331-37832.png)
|PR|·|QR|=
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103331-92794.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103331-63967.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103331-14505.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103331-14505.png)
=(x1+
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103331-14505.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103331-14505.png)
=(1+k2) x1 x2+(
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103331-14505.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103332-51328.png)
= -4(1+k2)+4k(
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103331-14505.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103332-51328.png)
=4(k2+
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103332-83962.png)
∵k2+
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103332-83962.png)
又α∈[
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103332-79520.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103332-90593.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103332-26477.png)
从而|PR|·|QR|的最小值为16. ………12分
解法二:(I)同解法一.
(Ⅱ)设直线l2的方程为y=kx+1,
把直线方程与抛物线方程联立消去y得 x2-4kx-4=0.
记P(x1,y1),Q(x2,y2),则
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103331-46554.png)
PR|·|QR|=
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103333-29144.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103333-29144.png)
=(1+k2)·(x1+
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103331-14505.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103331-14505.png)
下同解法一.
核心考点
试题【(本小题满分12分)已知定点,直线交轴于点,记过点且与直线相切的圆的圆心为点.(I)求动点的轨迹的方程;(Ⅱ)设倾斜角为的直线过点,交轨迹于两点 ,交直线于点.】;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/20191024/20191024103323-95434.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103323-77989.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103323-20559.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103323-16101.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103323-85649.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103323-92494.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103323-92491.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103323-72474.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103323-81479.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103320-95498.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103320-70202.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103318-11885.png)
A.x=8 | B.x=-8 | C.x=4 | D.x=-4 |
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024103313-24825.png)
△OAB的面积为( )
A.![]() | B.![]() | C.![]() | D.![]() |
A.2 | B.4 | C.8 | D.16 |
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