如果抛物线C1的顶点在抛物线C2上,抛物线C2的顶点也在抛物线C1上,那么我们称抛物线C1与C2为“互相关联”的抛物线.如图,已知抛物线
与
是“互相关联”的抛物线,点A,B分别是抛物线C1,C2的顶点,抛物线C2经过点D(6,-1).
![](https://img.xkw.com/dksih/QBM/2020/4/19/2444740726415360/2445311166644224/STEM/1a4a5b6cd1644c2582dbced6d983dcc8.png?resizew=160)
(1)直接写出点A,B的坐标和抛物线C2的解析式.
(2)抛物线C2上是否存在点E,使得△ABE是以AB为直角边的直角三角形?如果存在,请求出点E的坐标;如果不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c735ccdd6bcec9b78ac53d74f2837185.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92900979222987dacb7facf45031a79e.png)
![](https://img.xkw.com/dksih/QBM/2020/4/19/2444740726415360/2445311166644224/STEM/1a4a5b6cd1644c2582dbced6d983dcc8.png?resizew=160)
(1)直接写出点A,B的坐标和抛物线C2的解析式.
(2)抛物线C2上是否存在点E,使得△ABE是以AB为直角边的直角三角形?如果存在,请求出点E的坐标;如果不存在,请说明理由.
更新时间:2020-04-20 06:21:12
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【推荐1】如图,平面直角坐标系中,O为坐标原点,抛物线
经过
,
两点,与y轴交于点D,与x轴交于另一点B.点E坐标为
,点P是线段BO上的一个动点,从点B开始以1个单位每秒的速度沿BO向终点O运动;
![](https://img.xkw.com/dksih/QBM/2015/5/14/1573839969886208/1573839976243200/STEM/7989d36bce4e4bdfb17593d7c47e06f8.png)
(1)求此抛物线的解析式;
(2)设运动时间为t秒,直线PE扫过四边形ABCD的面积为S,当直线PE与线段BC有交点时,求S关于t的函数关系式;
(3)能否将△OEB绕平面内某点旋转90°后使得△OEB的两个顶点落在抛物线上?若能,请直接写出旋转中心的坐标;若不能,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ecf1ef383d6dc9f6e61f00f12ed1af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ac70ef96add56f2ea2244fdb4a02d25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4cf7823dff40bfab291e89008c6bd78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b802c0b075f8ce21a494dde744dd0e90.png)
![](https://img.xkw.com/dksih/QBM/2015/5/14/1573839969886208/1573839976243200/STEM/7989d36bce4e4bdfb17593d7c47e06f8.png)
(1)求此抛物线的解析式;
(2)设运动时间为t秒,直线PE扫过四边形ABCD的面积为S,当直线PE与线段BC有交点时,求S关于t的函数关系式;
(3)能否将△OEB绕平面内某点旋转90°后使得△OEB的两个顶点落在抛物线上?若能,请直接写出旋转中心的坐标;若不能,请说明理由.
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【推荐2】在平面直角坐标系
中,抛物线
与
轴交于点
,
,与
轴交于点
,对称轴直线
与
轴交于点
,连接
,
,
的面积为
.
(2)如图2,点
为抛物线上的一个动点,点
的横坐标为
,过点
作直线
轴交直线
于点
,
于点
,当
时,求
的值;
(3)抛物线
与
关于
轴对称,若点
是抛物线
上一点,点
在直线
上,点
在坐标平面内,当四边形
是正方形时,请求出点
的横坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a90385c676848de67293e3ed6bc000fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00b43a40d1c78f48590b69efdfb7337e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6570863c11e0c2a0dddc9ebb623e5c85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/661ff55b5ebbadfb600989af3cfce2fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ad9de54606c4a72d0f8decb6b0e63b2.png)
(2)如图2,点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37a11aa6595e26b877f3096a087e1ec5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecf783d3172debc69f963495e5bff78d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c1ec8da5eb4b009b2d4f12777cacfec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b7196cf68ef46f81af895e0cacadbf0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a90385c676848de67293e3ed6bc000fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e94556fc78e40aef7e9a3212f1ae24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e94556fc78e40aef7e9a3212f1ae24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9d0e5808a10c2e8823ace3267f5ed7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
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【推荐1】如图,Rt△ABO的两直角边OA、OB分别在x轴的负半轴和y轴的正半轴上,O为坐标原点,A、B两点的坐标分别为(﹣3,0)、(0,4),抛物线y=x2+bx+c经过点B,且顶点在直线x=上.
![](https://img.xkw.com/dksih/QBM/2014/11/21/1573758358577152/1573758364368896/STEM/ffb1ca666b41492396334adfe819c656.png)
(1)求抛物线对应的函数关系式;
(2)若把△ABO沿x轴向右平移得到△DCE,点A、B、O的对应点分别是D、C、E,当四边形ABCD是菱形时,试判断点C和点D是否在该抛物线上,并说明理由;
(3)在(2)的条件下,连接BD,已知对称轴上存在一点P使得△PBD的周长最小,求出P点的坐标;
![](https://img.xkw.com/dksih/QBM/2014/11/21/1573758358577152/1573758364368896/STEM/ffb1ca666b41492396334adfe819c656.png)
(1)求抛物线对应的函数关系式;
(2)若把△ABO沿x轴向右平移得到△DCE,点A、B、O的对应点分别是D、C、E,当四边形ABCD是菱形时,试判断点C和点D是否在该抛物线上,并说明理由;
(3)在(2)的条件下,连接BD,已知对称轴上存在一点P使得△PBD的周长最小,求出P点的坐标;
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名校
解题方法
【推荐2】问题提出
(1)如图1,在
中,
.请在
内画一个正方形,使得这个正方形一个内角为
,其余顶点落在
的边上;
问题探究
(2)如图,
为一块锐角三角形木板,其中
.
①如图2,若要在
中做出一个正方形,使正方形边落在
上,另外两个顶点分别落在
上,则该正方形的面积为____________.
②如图3,若要在
中做出一个平行四边形,使平行四边形一边
落在
上,另两顶点落在
上,请求出满足条件的平行四边形面积的最大值.
问题解决
(3)如图4有一四边形
与
交于
,现要在
中截出平行四边形
,使得平行四边形一边
与
平行,四个顶点
落在
的四边上,当
时,
____________.
(1)如图1,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f8f88798ec42a58dccd212586382b23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed10df4140819d5451773a45de66201b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/194741f4d2ae7ee44cafca780361446a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
问题探究
(2)如图,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccb31aced57fab505d1dd215fcb31400.png)
①如图2,若要在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dec2ca6438c82b43f746057d8129885.png)
②如图3,若要在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dec2ca6438c82b43f746057d8129885.png)
问题解决
(3)如图4有一四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34d94651fcfd4fe8d51fdcdc9ab8a215.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e3d4815b230ab49048cf843dfaf62e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/611f100dcfa7803db6eb233e2e7f2dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42c2d86d8daea5e652d99fe1c6bc3f9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e5ed3d3c8904b2f55b0d354845ff111.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a71a9d21f77e9535de152bb33f802bb.png)
![](https://img.xkw.com/dksih/QBM/2021/6/20/2746979978895360/2801027067584512/STEM/ef30f57d-83b0-4576-9345-5aaf2edea7a7.png?resizew=614)
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【推荐3】定义:由两条与x轴有着相同的交点,并且开口方向相同的抛物线所围成的封闭曲线称为“月牙线”.如图,抛物线C1与抛物线C2组成一个开口向上的“月牙线”,抛物线C1与抛物线C2与x轴有相同的交点M,N(点M在点N的左侧),与y轴的交点分别为A,B且点A的坐标为(0,﹣3),抛物线C2的解析式为y=mx2+4mx﹣12m,(m>0).
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/20/b6f54206-f356-4644-9e28-2ffcf0bb6329.png?resizew=208)
(1)请你根据“月牙线”的定义,设计一个开口向下.“月牙线”,直接写出两条抛物线的解析式;
(2)求M,N两点的坐标;
(3)在第三象限内的抛物线C1上是否存在一点P,使得△PAM的面积最大?若存在,求出△PAM的面积的最大值;若不存在,说明理由.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/20/b6f54206-f356-4644-9e28-2ffcf0bb6329.png?resizew=208)
(1)请你根据“月牙线”的定义,设计一个开口向下.“月牙线”,直接写出两条抛物线的解析式;
(2)求M,N两点的坐标;
(3)在第三象限内的抛物线C1上是否存在一点P,使得△PAM的面积最大?若存在,求出△PAM的面积的最大值;若不存在,说明理由.
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