题目
题型:海南省中考真题难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173020-83091.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173020-66075.gif)
(2)设P(x,y)是(1)所得抛物线上的一个动点,过点P作直线l⊥x轴于点M,交直线BC于点N。
①若点P在第一象限内,试问:线段PN的长度是否存在最大值?若存在,求出它的最大值及此时x的值;若不存在,请说明理由;
②求以BC为底边的等腰△BPC的面积。
答案
令y=0得x=3;
令x=0,得y=3,
∴B(3,0),C(0,3),
∵点B、C在抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173020-74335.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173021-56217.gif)
解得b=2,c=3,
∴所求函数关系式为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173021-56113.gif)
(2)①∵点P(x,y)在抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173021-21639.gif)
且PN⊥x轴,
∴设点P的坐标为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173021-15472.gif)
同理可设点N的坐标为(x,-x+3),
又点P在第一象限,
∴PN=PM-NM
=(
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173021-18267.gif)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173022-79795.gif)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173022-47227.gif)
∴当x=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173022-79761.gif)
线段PN的长度的最大值为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173022-34594.gif)
②由题意知,点P在线段BC的垂直平分线上,
又由①知,OB=OC,
∴BC的中垂线同时也是∠BOC的平分线,
∴设点P的坐标为(a,a),
又点P在抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173022-29124.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173023-57954.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173023-84316.gif)
解得:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173023-20971.gif)
∴点P的坐标为:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173023-73424.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173024-30107.gif)
若点P的坐标为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173024-48270.gif)
此时点P在第一象限,在Rt△OMP和Rt△BOC中,
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173024-36857.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173024-48586.gif)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173024-76202.gif)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173025-80191.gif)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173025-51112.gif)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173025-75931.gif)
若点P的坐标为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173025-54311.gif)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173026-91117.gif)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173026-67078.gif)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173026-27911.gif)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173026-98574.gif)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173026-23134.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173027-11003.gif)
核心考点
试题【如图,在平面直角坐标系中,直线y=-x+3与x轴、y轴分别交于点B、C;抛物线经过B、C两点,并与x轴交于另一点A。(1)求该抛物线所对应的函数关系式;(2)设】;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/20191020/20191020173001-31699.gif)
(2)当△ADP是直角三角形时,求点P的坐标;
(3)在问题(2)的结论下,若点E在x轴上,点F在抛物线上,问是否存在以A、P、E、F为顶点的平行四边形?若存在,求点F的坐标;若不存在,请说明理由。