题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070029-89961.png)
(1)求抛物线的解析式及顶点B的坐标;
(2)求证:CB是△ABE外接圆的切线;
(3)试探究坐标轴上是否存在一点P,使以D、E、P为顶点的三角形与△ABE相似,若存在,直接写出点P的坐标;若不存在,请说明理由;
(4)设△AOE沿x轴正方向平移t个单位长度(0<t≤3)时,△AOE与△ABE重叠部分的面积为s,求s与t之间的函数关系式,并指出t的取值范围.
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070029-85188.png)
答案
(1)y=a(x﹣3)(x+1);点B(1,4)
(2)见解析
(3)见解析
(4)s=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070029-56795.png)
解析
(1)由题意,设抛物线解析式为y=a(x﹣3)(x+1).
将E(0,3)代入上式,解得:a=﹣1.
∴y=﹣x2+2x+3.
则点B(1,4).
(2)证明:如图1,过点B作BM⊥y于点M,则M(0,4).
在Rt△AOE中,OA=OE=3,
∴∠1=∠2=45°,AE=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070030-85081.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070030-93851.png)
在Rt△EMB中,EM=OM﹣OE=1=BM,
∴∠MEB=∠MBE=45°,BE=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070030-17643.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070030-31983.png)
∴∠BEA=180°﹣∠1﹣∠MEB=90°.
∴AB是△ABE外接圆的直径.
在Rt△ABE中,tan∠BAE=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070031-62292.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070031-89566.png)
∴∠BAE=∠CBE.
在Rt△ABE中,∠BAE+∠3=90°,∴∠CBE+∠3=90°.
∴∠CBA=90°,即CB⊥AB.
∴CB是△ABE外接圆的切线.
(3)解:Rt△ABE中,∠AEB=90°,tan∠BAE=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070031-89566.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070031-16760.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070031-30984.png)
若以D、E、P为顶点的三角形与△ABE相似,则△DEP必为直角三角形;
①DE为斜边时,P1在x轴上,此时P1与O重合;
由D(﹣1,0)、E(0,3),得OD=1、OE=3,即tan∠DEO=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070031-89566.png)
满足△DEO∽△BAE的条件,因此 O点是符合条件的P1点,坐标为(0,0).
②DE为短直角边时,P2在x轴上;
若以D、E、P为顶点的三角形与△ABE相似,则∠DEP2=∠AEB=90°,sin∠DP2E=sin∠BAE=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070031-82968.png)
而DE=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070032-11476.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070032-22146.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070032-22146.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070031-82968.png)
即:P2(9,0);
③DE为长直角边时,点P3在y轴上;
若以D、E、P为顶点的三角形与△ABE相似,则∠EDP3=∠AEB=90°,cos∠DEP3=cos∠BAE=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070032-56207.png)
则EP3=DE÷cos∠DEP3=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070032-89865.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070033-37818.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070033-20358.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070033-27375.png)
综上,得:P1(0,0),P2(9,0),P3(0,﹣
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070033-27375.png)
(4)解:设直线AB的解析式为y=kx+b.
将A(3,0),B(1,4)代入,得
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070033-19824.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070034-45176.png)
∴y=﹣2x+6.
过点E作射线EF∥x轴交AB于点F,当y=3时,得x=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070034-20572.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070034-20572.png)
情况一:如图2,当0<t≤
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070034-20572.png)
则ON=AD=t,过点H作LK⊥x轴于点K,交EF于点L.
由△AHD∽△FHM,得
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070034-55999.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070034-28204.png)
解得HK=2t.
∴S阴=S△MND﹣S△GNA﹣S△HAD=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070034-57689.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070034-57689.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070035-95013.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070035-28532.png)
情况二:如图3,当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070035-28532.png)
由△IQA∽△IPF,得
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070035-11730.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070036-90243.png)
解得IQ=2(3﹣t).
∴S阴=S△IQA﹣S△VQA=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070035-95013.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070036-55556.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070036-55556.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070036-55556.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070036-95966.png)
综上所述:s=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070029-56795.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070036-80205.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070037-40744.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070037-52762.png)
核心考点
试题【如图甲,四边形OABC的边OA、OC分别在x轴、y轴的正半轴上,顶点在B点的抛物线交x轴于点A、D,交y轴于点E,连接AB、AE、BE.已知tan∠CBE=,A】;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
A.(6,3) | B.(6,5) | C.(-4,3) | D.(-4,5) |
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070019-47895.jpg)
(1)求抛物线的解析式;
(2)设抛物线的对称轴与x轴交于点M,问在对称轴上是否存在点P,使△CMP为等腰三角形?若存在,请直接写出所有符合条件的点P的坐标;若不存在,请说明理由.
(3)如图②,若点E为第二象限抛物线上一动点,连接BE、CE,求四边形BOCE面积的最大值,并求此时的点E的坐标.
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070013-60479.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070014-97613.png)
给出五种说法:①ab<0;
②方程
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070014-13855.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070014-27432.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070014-18361.png)
③a+b+c>0;④当x>1时,y随x的增大而增大;
⑤当y>0时,-1<x<3.
其中,正确的说法有
(把你认为正确的说法的序号都填上).
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070003-79171.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070004-50146.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070004-46893.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070004-36608.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070004-65327.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070004-29437.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019070005-60232.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065944-75034.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065944-13718.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065944-70721.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065944-65317.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065945-68769.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065945-89168.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065944-75034.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065945-16895.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065944-70721.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065944-65317.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065945-66307.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065944-13718.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065946-89207.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065946-45832.png)
(1) 填空:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065946-61978.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065946-12913.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065946-89207.png)
(2)设抛物线与
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065946-89696.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065947-24594.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065944-75034.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065947-75325.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065946-89696.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065947-10472.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065944-75034.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065947-24594.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065947-62506.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065947-24594.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065947-62506.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065946-89207.png)
(3)在(2)的条件下,若
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065947-62772.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065948-16834.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065948-52997.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065948-36240.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065948-95071.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065947-75325.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065949-55884.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065948-36240.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065949-26304.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065948-52997.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019065949-93643.png)
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