题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065442-91991.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065443-78358.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065443-35584.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065443-82379.png)
(1)求此抛物线的解析式;
(2)设E是线段AB上的动点,作EF//AC交BC于F,连接CE,当△CEF的面积是△BEF面积的2倍时,求E点的坐标;
(3)若P为抛物线上A、C两点间的一个动点,过P作
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065443-35584.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065443-79955.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065444-56131.png)
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065442-91991.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065443-78358.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065444-33265.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065444-82055.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065444-43898.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065444-68555.png)
故所求二次函数的解析式为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065443-79955.png)
(2)∵S△CEF="2" S△BEF, ∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065445-19789.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065445-32039.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065445-62100.png)
∴△BEF~△BAC,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065445-92550.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065445-62741.png)
故E点的坐标为(
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065444-56131.png)
(3)解法一:由抛物线与
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065443-35584.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065446-17218.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065446-17218.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065446-69097.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065446-49630.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065446-27934.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065447-30178.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065446-69097.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065447-13443.png)
若设
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065447-80508.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065447-50415.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065447-38190.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065447-80508.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065443-35584.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065446-69097.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065447-38190.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065448-47272.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065448-11495.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065448-19149.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065448-19898.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065449-52627.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065449-55281.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065447-80508.png)
解法二:延长
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065449-55281.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065443-78358.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065449-51981.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065449-44618.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065449-55281.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065449-33639.png)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065450-77566.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065450-26998.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065450-13594.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065450-33697.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065451-50263.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065451-85650.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065451-67596.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065451-18819.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065451-95439.png)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065452-89635.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065449-33639.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065449-55281.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065450-77566.png)
(1)将A、B的坐标代入抛物线的解析式中,即可求出待定系数的值;
(2)根据抛物线的解析式可得出C点的坐标,易证得△ABC是直角三角形,则EF⊥BC;△CEF和△BEF同高,则面积比等于底边比,由此可得出CF=2BF;易证得△BEF∽△BAC,根据相似三角形的性质,即可求得BE、AB的比例关系,由此可求出E点坐标;
(3)PQ的长实际是直线AC与抛物线的函数值的差,可设P点横坐标为m,用m表示出P、Q的纵坐标,然后可得出PQ的长与m的函数关系式,根据所得函数的性质即可求出PQ最大时,m的值,也就能求出此时P点的坐标.
核心考点
试题【如图9, 已知抛物线与轴交于A (-4,0) 和B(1,0)两点,与轴交于C点.(1)求此抛物线的解析式;(2)设E是线段AB上的动点,作EF//AC交BC于F】;主要考察你对二次函数定义等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065434-94082.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065434-37260.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065434-16934.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065435-43838.jpg)
(1)求该抛物线的解析式;
(2)在该抛物线的对称轴上存在一点
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065435-32558.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065435-64633.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065435-32558.png)
及
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065435-64633.png)
(3)在
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065435-63496.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065435-18596.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065435-19199.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065435-33619.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065435-12519.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065436-75480.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065436-76590.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065436-92388.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065436-88183.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065436-75419.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065436-92388.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065437-54402.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065437-89087.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065435-33619.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065436-75419.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065436-88183.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065437-76264.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065435-19199.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065436-88183.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065436-75419.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065437-85654.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065429-72939.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065429-45364.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065429-40846.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065416-95395.png)
(1)当t为何值时,点M与点Q相遇?
(2)填空:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065416-97632.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065417-57875.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065417-46731.png)
(3)当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065417-34851.png)
(4)在整个运动过程中,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065417-19269.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065418-83220.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065413-91111.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065414-92180.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065414-44190.png)
A.2个 | B.4个 | C.6个 | D.无数个 |
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065407-22703.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065407-98828.png)
(1)求点D的坐标;
(2)若抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065408-62013.png)
(3)设(2)中的抛物线的对称轴与直线OD交于点M,点P为坐标轴上一动点,以P、O、M为顶点的三角形与△OCD相似,求出点M的坐标和符合条件的点P的坐标.
(4)当(3)中符合条件的△POM面积最大时,过点O的直线
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065408-10904.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065408-10845.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065408-41251.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065408-10904.png)
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