题目
题型:福建省月考题难度:来源:
(1)求抛物线的解析式;
(2)在抛物线上求点M,使△MOB的面积是△AOB面积的3倍;
(3)连结OA,AB,在x轴下方的抛物线上是否存在点N,使△OBN与△OAB相似?若存在,求出N点的坐标;若不存在,说明理由.
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190257-46058.gif)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190257-40844.gif)
∵抛物线过原点,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190258-92944.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190258-29624.gif)
∴抛物线的解析式为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190258-91419.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190258-25487.gif)
(2)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190259-13413.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190259-38117.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190259-40350.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190259-74758.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190259-78931.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190300-81254.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190300-58657.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190300-92246.gif)
解之,得
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190300-17848.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190300-18332.gif)
∴满足条件的点有两个:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190301-38792.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190301-74993.gif)
(3)不存在.
由抛物线的对称性,知
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190301-80752.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190301-58272.gif)
如图,若
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190302-67378.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190302-71110.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190302-89721.gif)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190303-66215.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190303-30082.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190303-33764.gif)
∴直线
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190303-86492.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190304-56462.gif)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190304-87729.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190304-16887.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190304-34137.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190304-33788.gif)
过
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190305-47732.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190305-27081.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190305-53412.gif)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190305-39151.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190306-42569.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190306-56825.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190307-67269.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190307-72920.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190307-31583.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190307-46200.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190307-54174.gif)
同理,在对称轴左边的抛物线上也不存在符合条件的点.
所以在该抛物线上不存在点N,使
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190308-80241.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190308-38447.gif)
核心考点
试题【如图,抛物线的顶点为A(2,1),且经过原点O,与x轴的另一个交点为B.(1)求抛物线的解析式;(2)在抛物线上求点M,使△MOB的面积是△AOB面积的3倍;(】;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
(1)求点A,B的坐标,并画出一次函数y=x-3的图象;
(2)求二次函数的解析式及它的最小值.
(1)求证:四边形AEFD是平行四边形;
(2)设AE=x,四边形DEGF的面积为y,求y关于x的函数关系式。
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190235-42067.gif)
(1)求C点坐标及直线BC的解析式
(2)一抛物线经过B、C两点,且顶点落在x轴正半轴上,求该抛物线的解析式并画出函数图象
(3)现将直线BC绕B点旋转与抛物线相交与另一点P,请找出抛物线上所有满足到直线AB距离为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190226-71701.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190226-60578.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190212-11631.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190212-82956.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190212-29916.gif)
(1)求抛物线的解析式及其顶点D的坐标;
(2)设直线
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190213-47493.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190213-12954.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190213-64838.gif)
(3)过点B作x轴的垂线,交直线
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190213-42821.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190214-89237.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020190214-51932.gif)
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