解:(1)∵函數(shù)y=
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(x>0,k是常數(shù))的圖象經(jīng)過(guò)A(1,4)
∴k=1×4=4.
(2)由(1)可知y=
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,
∴mn=4
∵BD=m,OD=n
∴
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m(4-n)=4
即4m-mn=8
∴m=3,n=
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即點(diǎn)B的坐標(biāo)為(3,
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).
觀察圖片可知當(dāng)1<x<3時(shí),
直線AB的圖象在反比例函數(shù)y=
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圖象的上方.
分析:(1)根據(jù)反比例函數(shù)上的點(diǎn)的坐標(biāo),橫縱坐標(biāo)的乘積相等,等于k求解即可;
(2)根據(jù)反比例函數(shù)上點(diǎn)的特點(diǎn)可求得mn=4,再根據(jù)△ABC的面積為4,求得m,n之間的另一個(gè)關(guān)系式,代入mn值即可求得m值,從而求出n值,得到點(diǎn)B的坐標(biāo).
點(diǎn)評(píng):此題綜合考查了反比例函數(shù)的性質(zhì),綜合性比較強(qiáng),注意反比例函數(shù)上的點(diǎn)向x軸y軸引垂線形成的矩形面積(反比例函數(shù)上的點(diǎn)的坐標(biāo),橫縱坐標(biāo)的乘積相等)等于反比例函數(shù)的k值.