题目
题型:四川省中考真题难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144207-65630.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144208-49215.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144208-21470.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144208-55061.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144208-26331.png)
求抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144208-79108.png)
(1)求抛物线的解析式
(2)试判断直线CM与以AB为直径的圆的位置关
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144209-97898.png)
(3)在抛物线上是否存在点N,使得
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144209-23015.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144209-13298.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144209-70203.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144210-54356.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144210-57042.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144210-43362.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144211-84984.png)
又∵
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144211-89934.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144211-18335.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144211-52823.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144211-50999.png)
∵抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144211-61100.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144212-12608.png)
∴可设此抛物线为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144212-92201.png)
代入
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144212-15116.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144212-35670.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144213-46210.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144213-58470.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144213-39376.png)
(2)∵
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144213-46814.png)
∴以AB为直径的圆的直径为AB,
又∵
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144213-29794.png)
∴以AB为直径的圆的圆心为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144214-20554.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144214-89300.png)
∴此抛物线的顶点为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144214-97961.png)
又∵
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144214-76184.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144215-38515.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144215-19432.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144215-31245.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144215-54847.png)
∴直线CM与以AB为直径的圆相交。
(3)抛物线上存在点N,使得
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144216-88540.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144216-29144.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144216-67863.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144216-91838.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144217-89347.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144217-22219.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144217-48881.png)
于是可设与直线
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144217-32709.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144218-97203.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144218-38815.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144218-41045.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144218-50027.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144218-79815.png)
∴所设直线为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144219-92146.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144219-52636.png)
∴由
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144219-72016.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144219-97262.png)
∴由
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144220-78306.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144220-18226.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144220-55733.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144220-17682.png)
∴由
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144221-55041.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144221-75527.png)
故:物线上存在点N,使得
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144221-28236.png)
核心考点
试题【如图,已知点点C在y轴的正半轴上,且抛物线经过三点,其顶点为.求抛物线的解析式;(1)求抛物线的解析式(2)试判断直线CM与以AB为直径的圆的位置关系,并加以证】;主要考察你对二次函数的应用等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144200-10809.png)
(1)请你从所学习过的一次函数、反比例函数和二次函数中确定哪种函数能表示所用燃气量y升与旋钮角度x度的变化规律?说明确定是这种函数而不是其它函数的理由,并求出它的解析式;
(2)当旋钮角度为多少时,烧开一壶水所用燃气量最少?最少是多少?
(3)某家庭使用此款燃气灶,以前习惯把燃气开到最大,现采用最节省燃气的旋钮角度,每月平均能节
约燃气10立方米,求该家庭以前每月的平均燃气用量.
(2)当旋钮角度为多少时,烧开一壶水所用燃气量最少?最少是多少?
(3)某家庭使用此款燃气灶,以前习惯把燃气开到最大,现采用最节省燃气的旋钮角度,每月平均能节
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144200-49673.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144200-89636.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144150-74052.png)
(1).求抛物线的解析式.
(2)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144150-23090.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144151-35897.png)
注:二次函数y=ax2+bx+c(a≠0)的对称轴是直线x=-
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144151-93382.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144151-46301.png)
如图,已知抛物线与坐标轴分别交于A(﹣2 ,0)、B(2 ,0)、C(0 ,﹣1)三点,过坐标原点O的直线y=kx与抛物线
交于M、N两点.分别过点C,D(0,﹣2)作平行于x轴的直线l1、l2.
(1)求抛物线对应二次函数的解析式;
(2)求证以ON为直径的圆与直线l1相切;
(3)求线段MN的长(用k表示),并证明M、N两点到直线l2的距离之和等于线段MN的长.
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144138-51351.png)
(1)求抛物线对应二次函数的解析式;
(2)求证以ON为直径的圆与直线l1相切;
(3)求线段MN的长(用k表示),并证明M、N两点到直线l2的距离之和等于线段MN的长.
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144138-16912.png)
(1)求c的值;
(2)若a=﹣1,且抛物线与矩形有且只有三个交点A、D、E,求△ADE的面积S的最大值;
(3)若抛物线与矩形有且只有三个交点A、M、N,线段MN的垂直平分线l过点0,交线段BC于点F.当BF=1时,求抛物线的解析式.
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144120-72013.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020144117-39754.png)
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