已知实数a,b,c,满足a>0,a^2-2ab+c^2=0,bc>a^2 (1)求证:b>c>0 (2)试确定实数a,c的大小关系
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(1) a>0, a^2-2ab+c^2=0 => b>0 ,bc>a^2>0 => c>0
0=a^2-2ab+c^2=(a-b)^2+c^2-b^2 => b^2-c^2=(a-b)^2≥0
=> b≥c>0
若b=c,则b^2-c^2=(a-b)^2=0 => a=b
=> a=b=c =>bc=a^2 与题意bc>a^2矛盾
从而,b>c>0
(2)a^2<bc<b^2 => a<b
c^2=2ab-a^2>2a^2-a^2=a^2 => c>a
从而b>c>a>0
0=a^2-2ab+c^2=(a-b)^2+c^2-b^2 => b^2-c^2=(a-b)^2≥0
=> b≥c>0
若b=c,则b^2-c^2=(a-b)^2=0 => a=b
=> a=b=c =>bc=a^2 与题意bc>a^2矛盾
从而,b>c>0
(2)a^2<bc<b^2 => a<b
c^2=2ab-a^2>2a^2-a^2=a^2 => c>a
从而b>c>a>0
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