题目
题型:山东省中考真题难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141305-36587.png)
(1)求该抛物线的解析式.
(2)若点P是AB上的一动点,过点P作PE∥AC,交BC于E,连接CP,求△PCE面积的最大值.
(3)若点D为OA的中点,点M是线段AC上一点,且△OMD为等腰三角形,求M点的坐标.
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141305-74510.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141305-60404.png)
得
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141305-48771.png)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141306-33736.png)
∴该抛物线的解析式为y=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141306-67910.png)
(2)令y=0,即
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141306-54586.png)
∴A(﹣4,0),S△ABC=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141306-58102.png)
∵PE∥AC,
∴∠BPE=∠BAC,∠BEP=∠BCA,
∴△PBE∽△ABC,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141307-34569.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141307-89320.png)
化简得:S△PBE=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141307-79101.png)
S△PCE=S△PCB﹣S△PBE=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141308-37491.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141308-11240.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141308-24408.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141308-35528.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141309-92945.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141309-36318.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141309-35611.png)
∴当x=﹣1时,S△PCE的最大值为3.
(3)△OMD为等腰三角形,可能有三种情形:
(I)当DM=DO时,如答图①所示. DO=DM=DA=2,
∴∠OAC=∠AMD=45°,
∴∠ADM=90°,
∴M点的坐标为(﹣2,﹣2);
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141309-71561.png)
(II)当MD=MO时,如答图②所示.
过点M作MN⊥OD于点N,则点N为OD的中点,
∴DN=ON=1,AN=AD+DN=3,
又△AMN为等腰直角三角形,
∴MN=AN=3,
∴M点的坐标为(﹣1,﹣3);
(III)当OD=OM时,
∵△OAC为等腰直角三角形,
∴点O到AC的距离为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141310-50007.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141310-10245.png)
即AC上的点与点O之间的最小距离为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141310-37509.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141310-41735.png)
∴OD=OM的情况不存在.综上所述,点M的坐标为(﹣2,﹣2)或(﹣1,﹣3).
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141311-57343.png)
核心考点
试题【如图,抛物线y= x2+bx+c与y轴交于点C(0,﹣4),与x轴交于点A,B,且B点的坐标为(2,0)(1)求该抛物线的解析式.(2)若点P是AB上的一动点,】;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/20191020/20191020141252-10828.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141253-96246.png)
(1)求这个一次函数的解析式;
(2)求二次函数的解析式;
(3)如果点C在这个二次函数的图像上,且点C的横坐标为5,求tan∠CAB的值.
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141253-58891.png)
![魔方格](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020141246-62878.png)
(1)按照要求填表: