题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071555-76963.png)
小题1:求此抛物线的解析式
小题2:点P(x,y)为此抛物线上一动点,连接MP交此抛物线的对称轴于点D,当△DMN为直角三角形时,求点P的坐标;
小题3:设此抛物线与y轴交于点C,在此抛物线上是否存在点Q,使∠QMN=∠CNM ?若存在,求出点Q的坐标;若不存在,说明理由.
答案
小题1:∵
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071555-15260.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071555-61840.png)
由题意,得M(
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071556-93798.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071556-16701.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071556-68464.png)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071556-47638.png)
∴此抛物线的解析式为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071557-27932.png)
小题2:设抛物线的对称轴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071557-17988.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071557-41642.png)
若△DMN为直角三角形,则
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071557-37377.png)
∴D1(
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071557-40131.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071558-90505.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071558-91749.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071557-40131.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071558-33196.png)
直线MD1为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071558-38063.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071559-88499.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071559-91894.png)
将P(x,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071559-73367.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071559-88499.png)
得
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071559-54457.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071559-21694.png)
解①得
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071600-59226.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071600-82979.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071600-74042.png)
解②得
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071600-24225.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071601-73024.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071601-78144.png)
小题3:设存在点Q(x,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071559-73367.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071601-65548.png)
使得∠QMN=∠CNM.
①若点Q在MN上方,过点Q作QH⊥MN,
交MN于点H,则
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071601-47954.png)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071601-79146.png)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071602-94363.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071600-82979.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071602-24918.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071558-90505.png)
②若点Q在MN下方,
同理可得
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071602-84407.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071602-56203.png)
解析
(2)设抛物线的对称轴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071557-17988.png)
(3)先设存在点Q(x,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071559-73367.png)
核心考点
试题【在平面直角坐标系xOy中,抛物线经过点N(2,-5),过点N作x轴的平行线交此抛物线左侧于点M,MN=6.小题1:求此抛物线的解析式小题2:点P(x,y)为此抛】;主要考察你对二次函数定义等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071547-21694.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071547-54089.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071547-60151.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071547-40787.png)
(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071547-34502.png)
(2)求证:无论
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071547-34502.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071547-67870.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071547-67870.png)
(3)已知抛物线与
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071547-67870.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071547-67870.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071547-67870.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071547-67870.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071547-67870.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071547-21694.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071535-45031.png)
(1)证明:不论k取何值,抛物线L的顶点C总在抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071535-52331.png)
(2)已知
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071535-29104.png)
(3)在(2)A、B间距取得最大值条件下(点A在点B的右侧),直线y=ax+b是经过点A,且与抛物线L相交于点D的直线. 问是否存在点D,使△ABD为等边三角形,如果存在,请写出此时直线AD的解析式;如果不存在,请说明理由.
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071520-25584.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071520-67916.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071521-11756.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071521-82090.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071521-45823.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071521-91971.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071521-97009.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071522-94527.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071522-20064.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071522-40330.jpg)
(1)写出点C的坐标;
(2)求出抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071520-67916.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071522-67359.png)
(3)点
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071522-32929.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071523-78729.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071522-32929.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071511-59783.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071511-30482.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019071511-30035.png)
A.y1,y2开口方向,开口大小不一定相同 . |
B.y1,y2的对称轴相同. |
C.如果y1与x轴有两个不同的交点,则y2与x轴也有两个不同的交点. |
D.如果y2的最大值为m,则y1的最大值为km. |
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