题目
题型:贵州省月考题难度:来源:
![](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。
答案
设
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173001-26574.gif)
将C(0,3)代入上式,得
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173001-25035.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173001-25400.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173002-33347.gif)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173002-86987.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173002-20321.gif)
①当点P1为直角顶点时,点P1与点B重合(如图)
令y=0,得
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173003-95989.gif)
解得:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173003-56591.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173003-31281.gif)
∵点A在点B的右边,
∴B(1,0),A(3,0)
∴P1(1,0)
②当点A为△APD2的直角顶点是(如图)
∵OA=OC,∠AOC=90°,
∴∠OAD2=45°
当∠D2AP2=90°时,∠OAP2=45°,
∴AO平分∠D2AP2
又∵P2D2∥y轴,
∴P2D2⊥AO,
∴P2、D2关于x轴对称
设直线AC的函数关系式为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173003-17974.gif)
将A(3,0), C(0,3)代入上式得
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173003-82003.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173004-41222.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173004-19272.gif)
∵D2在
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173004-10963.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173004-38977.gif)
∴设D2(x,-x+3),P2(x,
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173005-77123.gif)
∴(
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173005-73409.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173005-67109.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173006-77167.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173006-30783.gif)
∴当x=2时,
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173006-92130.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173006-53204.gif)
∴P2的坐标为P2(2,-1)(即为抛物线顶点)
∴P点坐标为P1(1,0),P2(2,-1)。
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173007-45782.gif)
当点P的坐标为P2(2,-1)(即顶点Q)时,平移直线AP(如图)交x轴于点E,交抛物线于点F
当AP=FE时,四边形PAFE是平行四边形
∵P(2,-1),
∴可令F(x,1)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173007-88719.gif)
解之得:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173007-35664.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173007-30203.gif)
∴F点有两点,即F1(
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173008-81626.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173008-76344.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020173008-53474.gif)
核心考点
试题【如图,已知抛物线y=ax2+bx+c(a≠0)的顶点坐标为Q(2,-1),且与y轴交于点C(0,3),与x轴交于A,B两点(点A在点B的右侧),点P是该抛物线上】;主要考察你对二次函数的应用等知识点的理解。[详细]
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