若x,y,z均不为0,且x+y+z=0,证明:√(1/x^2+1/y^2+1/z^2)=|1/x+1/y+1/z|
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![若x,y,z均不为0,且x+y+z=0,证明:√(1/x^2+1/y^2+1/z^2)=|1/x+1/y+1/z|](/uploads/image/z/12049959-39-9.jpg?t=%E8%8B%A5x%2Cy%2Cz%E5%9D%87%E4%B8%8D%E4%B8%BA0%2C%E4%B8%94x%2By%2Bz%3D0%2C%E8%AF%81%E6%98%8E%EF%BC%9A%E2%88%9A%EF%BC%881%2Fx%5E2%2B1%2Fy%5E2%2B1%2Fz%5E2%29%3D%7C1%2Fx%2B1%2Fy%2B1%2Fz%7C)
若x,y,z均不为0,且x+y+z=0,证明:√(1/x^2+1/y^2+1/z^2)=|1/x+1/y+1/z|
若x,y,z均不为0,且x+y+z=0,证明:√(1/x^2+1/y^2+1/z^2)=|1/x+1/y+1/z|
若x,y,z均不为0,且x+y+z=0,证明:√(1/x^2+1/y^2+1/z^2)=|1/x+1/y+1/z|
证明:|1/x+1/y+1/z|^2=1/x^2+1/y^2+1/z^2+2/(xy)+2/(yz)+2/(zx)
=1/x^2+1/y^2+1/z^2+2/(xy)+2/z(1/x+1/y)
=1/x^2+1/y^2+1/z^2+2/(xy)+2/z[(x+y)/xy]
因为z=-x-y
:|1/x+1/y+1/z|^2=1/x^2+1/y^2+1/z^2+2/(xy)-2/(x+y)[(x+y)/xy]
=1/x^2+1/y^2+1/z^2
两边开根号,得到结论
证毕
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