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 |