说明:
抛物线与几何图形相结合的题目是考试的常考题型,同时也是考试的重点和难点.常见的题型有抛物线与平行四边形的结合、抛物线与等腰三角形的结合、抛物线与直角三角形的结合、抛物线与相似三角形的结合等等,其涉及到的知识点较多,知识点之间的综合性较强,故考生在平时应多给与关注,进行适量的练习,以期掌握这类题型的一般解决方法.
抛物线与直角三角形的结合
首先补充两个重要的知识点:
(1)直角三角形的性质: 直角三角形斜边上的中线等于斜边的一半.
(2)对于两条直线:
若,则.
注意 此结论通常用来求一次函数的解析式.
例如:直线的解析式为,直线与垂直,且直线经过点,求直线的解析式.
解:由题意可设直线为:
∵其图象经过点
∴
∴直线的解析式为.
▲例1.(2015.省实验中学)如图所示,抛物线与直线交于A、B两点,点A的纵坐标为,点B在轴上,直线AB与轴交于点F,点P是线段AB下方的抛物线上一动点,横坐标为,过点P作PC轴于C,交直线AB于D.
(1)求抛物线的解析式;
(2)当取何值时,线段PD的长度取得最大值,其最大值是多少?
(3)是否存在点P,使△PAD是直角三角形?若存在,求出点P的坐标;若不存在,说明理由.
提示: 要求会在平面直角坐标系中求一条线段的长度.
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▲例2.(2015.连云港)如图,已知一条直线过点( 0 , 4 ),且与抛物线交于A、B两点,其中点A的横坐标是.
(1)求这条直线的函数关系式及点B的坐标;
(2)在轴上是否存在点C,使得△ABC是直角三角形?若存在,求出点C的坐标,若不存在,请说明理由;
(3)过线段AB上一点P,作PM//轴,交抛物线于点M,点M在第一象限,点N为( 0 , 1),当点M的横坐标为何值时,MN+3MP的长度最大?最大值是多少?
提示: 要求会熟练使用勾股定理和两点间的距离公式.
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▲例3.如图,直线与抛物线相交于点A和点B,点P是线段AB上异于A、B的动点,过点P作PC轴于点D,交抛物线于点C.
(1)求抛物线的解析式;
(2)求△PAC为直角三角形时点P的坐标.
要点提示: 解决运动变化中的等腰三角形和直角三角形的存在性问题,首先要明确已知条件对图形的限定,然后进行分类讨论,分别画出符合要求的图形,再进行求解.
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▲例4.如图,在平面直角坐标系中,矩形OABC的三个顶点为O( 0 , 0 )、A( 0 , 8 )、C( 6 , 0 ),抛物线经过点A、C,与边AB交于点D.动点P从点C出发沿CB方向以每秒个单位的速度向点B运动,同时,点Q从点A出发沿AC方向以每秒1个单位的速度向点C运动,设运动时间为秒(),连结PQ.
(1)求抛物线的解析式;
(2)设△CPQ的面积为,求关于的函数关系式,并求为何值时,取得最大值;
(3)当△CPQ的面积取最大值时,在抛物线的对称轴上,是否存在点M,使△DMQ为直角三角形,若存在,请直接写出所有符合条件的点M的坐标;若不存在,请说明理由.
要点提示:在讨论直角三角形的存在性问题时,应分为三种情况:每个内角为直角.
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▲例5.如图,抛物线经过点,点B在抛物线上,CB//轴,且AB平分∠CAO.
(1)求抛物线的解析式;
(2)线段AB上有一动点P,过P作轴的平行线,交抛物线于点Q,求线段PQ的最大值;
(3)抛物线的对称轴上是否存在点M,使△ABM是以AB为直角边的直角三角形?如果存在,直接写出点M的坐标;如果不存在,说明理由.
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抛物线与等腰三角形的结合
当讨论一个等腰三角形的存在性问题时,也应分三种情况进行讨论:每条边作一次底边.
在解决问题时,往往会用到等腰三角形“三线合一”的性质以及勾股定理.
●例6.如图,在平面直角坐标系中,已知矩形ABCD的三个顶点B(4,0)、C(8,0)、D(8,8).抛物线过A、C两点.
(1)直接写出点A的坐标,并求出抛物线的解析式;
(2)动点P从点A出发,沿线段AB向终点B运动,同时点Q从点C出发,沿线段CD向终点D运动,速度均为每秒1个单位长度,运动时间为秒.过点P作PE⊥AB交AC于点E.
过点E作EF⊥AD于点F,交抛物线于点G.当为何值时,线段EG最长?
连结EQ.在点P、Q运动的过程中,判断有几个时刻使得△CEQ是等腰三角形?直接写出相应的值.
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●例7.如图,抛物线与轴交于A、B两点,与轴交于点C,抛物线的对称轴交轴于点D.已知A,C.
(1)求抛物线的表达式;
(2)在抛物线的对称轴上是否存在点P,使△PCD是以CD为腰的等腰三角形?如果存在,直接写出P点的坐标;如果不存在,请说明理由;
(3)点E是线段BC上的一个动点,过点E作轴的垂线与抛物线相交于点F,当点E运动到什么位置时,四边形CDBF的面积最大?求出四边形CDBF的最大面积及此时E点的坐标.
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●例8.如图,在平面直角坐标系中,抛物线与轴交于点A,B两点,直线与轴交于点D,与轴交于点C.点P是轴下方的抛物线上一动点,过点P作PF轴于点F,交直线CD于点E.设点P的横坐标为.
(1)求抛物线的解析式;
(2)若PE = 3EF,求的值;
(3)连结PC,是否存在点P,使△PCE为等腰直角三角形?若存在,请直接写出相应的点P的横坐标的值;若不存在,请说明理由.
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●例9.如图,抛物线交轴于点A、B,交轴于点C,直线过点A和点C.
(1)求抛物线的解析式;
(2)点P是该二次函数在第一象限的图象上一动点,连结AP、CP,求△PAC面积的最大值;
(3)设抛物线的对称轴与轴交于点M,在对称轴上是否存在点R,使△CMR为等腰三角形?若存在,请直接写出所有符合条件的点R的坐标;若不存在,请说明理由.
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●例10.如图,已知抛物线过点( 2 , 0 ),且顶点A的纵坐标为1.过抛物线上一点P向直线作垂线,垂足为点Q.点B在抛物线的对称轴上,点B的纵坐标为.直线与抛物线的对称轴交于点C.
(1)求抛物线的解析式;
(2)若BQ=PQ,求点P的坐标,并判断此时△PBQ是不是等边三角形,请说明理由;
(3)在抛物线上是否存在点P,使PQ=PB成立?若存在,请直接写出点P的坐标或点P的运动范围;若不存在,请说明理由.
抛物线与特殊四边形的结合