题目内容
(本题满分12分)如图,在平面直角坐标系中,直线分别交轴,轴于A,B两点,点C为OB的中点,点D在第二象限,且四边形AOCD为矩形.
(1)直接写出点A,B的坐标,并求直线AB与CD交点的坐标;
(2)动点P从点C出发,沿线段CD以每秒1个单位长度的速度向终点D运动;同时,动点M从点A出发,沿线段AB以每秒个单位长度的速度向终点B运动,过点P作,垂足为H,连接,.设点P的运动时间为秒.
①若△MPH与矩形AOCD重合部分的面积为1,求的值;
②点Q是点B关于点A的对称点,问是否有最小值,如果有,求出相应的点P的坐标;如果没有,请说明理由.
解:(1),.···································································· 1分
当时,,.
所以直线AB与CD交点的坐标为.···················································· 2分
(2)
当0<<时,△MPH与矩形AOCD重合部分的面积即△MPH的面积.
过点M作,垂足为N.
由△AMN∽△ABO,得.
∴.∴.········································································ 4分
∴△MPH的面积为.
当时,.············································································· 5分
当<≤3时,设MH与CD相交于点E,△MPH与矩形AOCD重合部分的面积即
△PEH的面积.
过点M作于G,交HP的延长线于点F.
.
.
由△HPE∽△HFM,得.
∴.∴.································································ 8分
∴△PEH的面积为.
当时,.
综上所述,若△MPH与矩形AOCD重合部分的面积为1,为1或.·················· 9分
(3)有最小值.
连接PB,CH,则四边形PHCB是平行四边形.
∴. ∴.
当点C,H,Q在同一直线上时,的值最小.···································· 11分
∵点C,Q的坐标分别为,, ∴直线CQ的解析式为,
∴点H的坐标为. 因此点P的坐标为.······························ 12分
解析:略
(本题满分12分)
如图,的顶点A、B在二次函数的图像上,又点A、B[来分别在轴和轴上,∠ABO=.
1.(1)求此二次函数的解析式;(4分)
2.
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点在上述函数图像上,当与相似时,求点的坐标.(8分)