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AB两点,点COB的中点,点D在第二象限,且四边形AOCD为矩形.

(1)直接写出点AB的坐标,并求直线ABCD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,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

时,

所以直线ABCD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,垂足为N

由△AMN∽△ABO,得

.∴.········································································ 4分

∴△MPH的面积为

时,.············································································· 5分

≤3时,设MHCD相交于点E,△MPH与矩形AOCD重合部分的面积即

PEH的面积.

过点MGHP的延长线于点F

由△HPE∽△HFM,得

.∴.································································ 8分

∴△PEH的面积为

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PBCH,则四边形PHCB是平行四边形.

.     ∴

当点CHQ在同一直线上时,的值最小.···································· 11分

∵点CQ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

 

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