题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044827-61476.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044827-70840.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044827-90217.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044827-94533.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044827-90236.png)
(1)求抛物线和直线的解析式;
(2)设点P是直线AC上一点,且
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044828-43123.png)
(3)若直线
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044828-39921.png)
①是否存在a的值,使得∠MON=900?若存在,求出a的值;若不存在,请说明理由;
②猜想当∠MON>900时,a的取值范围(不写过程,直接写结论).
(参考公式:在平面直角坐标系中,若M(x1,y1),N(x2,y2),则M,N两点间的距离为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044828-96773.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044828-59471.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044828-53345.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044828-27063.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044828-46519.png)
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044827-90217.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044827-70840.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044827-94533.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044828-60456.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044829-46153.png)
∴抛物线的解析式为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044829-84511.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044829-58910.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044829-86475.png)
令x=0,得
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044829-29152.png)
将A、C的坐标代入
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044827-61476.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044829-83415.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044829-22958.png)
∴直线AC的解析式为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044828-59471.png)
(2)分两种情况:
①点P在线段AC上时,过P作PH⊥x轴,垂足为H,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044830-12988.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044830-37329.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044830-62919.png)
∵PH∥CP,∴△APH∽△ACO。
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044830-59614.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044830-34987.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044830-41297.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044830-73728.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044830-96987.png)
②点P在线段CA的延长线上时,过P作PG⊥x轴,垂足为G,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044830-33998.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044830-37329.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044830-76626.png)
∵PG∥CO,∴△APG∽△ACO。
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044831-24163.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044831-85474.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044831-47466.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044831-14380.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044831-80008.png)
综上所述,点P的坐标为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044828-53345.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044828-27063.png)
(3)①存在。
假设存在a的值,使直线
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044828-39921.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044829-84511.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044831-37702.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044831-47504.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044831-19929.png)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044831-34476.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044832-96975.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044832-63274.png)
∵∠MON=900,∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044832-51539.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044832-43494.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044832-23731.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044832-87693.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044833-28005.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044833-72642.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044833-74169.png)
∴存在
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044833-72642.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044833-74169.png)
②当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044828-46519.png)
(1)根据当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044827-90217.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044827-70840.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044827-94533.png)
(2)分点P在线段AC上和两种情况讨论即可。
(3)①应用一元二次方程根与系数的关系和勾股定理求解。
②如图,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044833-47158.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044833-72642.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044833-74169.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044834-56057.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044834-87128.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044828-46519.png)
核心考点
试题【如图1所示,已知直线与x轴、y轴分别交于A、C两点,抛物线经过A、C两点,点B是抛物线与x轴的另一个交点,当时,y取最大值.(1)求抛物线和直线的解析式;(2)】;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/20191019/20191019044819-13132.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044819-76224.png)
A.-2 | B.-1 | C.1 | D.2 |
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044811-41846.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044811-39978.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044811-72587.png)
(1)试求b,c的值,并写出该二次函数表达式;
(2)动点P从A到D,同时动点Q从C到A都以每秒1个单位的速度运动,问:
①当P运动到何处时,有PQ⊥AC?
②当P运动到何处时,四边形PDCQ的面积最小?此时四边形PDCQ的面积是多少?
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044807-54125.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044807-73363.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044758-67162.png)
(1)求证:∠APE=∠CFP;
(2)设四边形CMPF的面积为S2,CF=x,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044758-49355.png)
①求y关于x的函数解析式和自变量x的取值范围,并求出y的最大值;
②当图中两块阴影部分图形关于点P成中心对称时,求y的值.
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019044754-30429.png)
(1)求抛物线的解析式;
(2)求抛物线的顶点坐标.
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