设a,b,c,x,y,z是实数,若a^2+b^2+c^2=25,x^2+y^2+z^2=36,ax+by+cz=30,求(2007a+5b+8c)/(2007x+5y+8z)的值.要过程,详细清楚的,能让我懂.
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![设a,b,c,x,y,z是实数,若a^2+b^2+c^2=25,x^2+y^2+z^2=36,ax+by+cz=30,求(2007a+5b+8c)/(2007x+5y+8z)的值.要过程,详细清楚的,能让我懂.](/uploads/image/z/10140244-52-4.jpg?t=%E8%AE%BEa%2Cb%2Cc%2Cx%2Cy%2Cz%E6%98%AF%E5%AE%9E%E6%95%B0%2C%E8%8B%A5a%5E2%2Bb%5E2%2Bc%5E2%3D25%2Cx%5E2%2By%5E2%2Bz%5E2%3D36%2Cax%2Bby%2Bcz%3D30%2C%E6%B1%82%282007a%2B5b%2B8c%29%2F%282007x%2B5y%2B8z%29%E7%9A%84%E5%80%BC.%E8%A6%81%E8%BF%87%E7%A8%8B%2C%E8%AF%A6%E7%BB%86%E6%B8%85%E6%A5%9A%E7%9A%84%2C%E8%83%BD%E8%AE%A9%E6%88%91%E6%87%82.)
设a,b,c,x,y,z是实数,若a^2+b^2+c^2=25,x^2+y^2+z^2=36,ax+by+cz=30,求(2007a+5b+8c)/(2007x+5y+8z)的值.要过程,详细清楚的,能让我懂.
设a,b,c,x,y,z是实数,若a^2+b^2+c^2=25,x^2+y^2+z^2=36,ax+by+cz=30,求(2007a+5b+8c)/(2007x+5y+8z)的值.
要过程,详细清楚的,能让我懂.
设a,b,c,x,y,z是实数,若a^2+b^2+c^2=25,x^2+y^2+z^2=36,ax+by+cz=30,求(2007a+5b+8c)/(2007x+5y+8z)的值.要过程,详细清楚的,能让我懂.
考虑关于t的二次函数
(a^2+b^2+c^2)^2*t^2-2(ax+by+cz)t+(x^2+y^2+z^2)=25t^2-60t+36=(5t-6)^2
而左边还可以写成(at-x)^2+(bt-y)^2+(ct-z)^2
所以,如果令t=1.2就得到(1.2a-x)^2+(1.2b-y)^2+(1.2c-z)^2=0
所以必有1.2a-x=1.2b-y=1.2c-z=0
于是(2007a+5b+8c)/(2007x+5y+8z)=1/1.2=5/6
由柯西不等式可知:(a^2+b^2+c^2)(x^2+y^2+z^2)≥(ax+by+cz)^2
当且仅当a/x=b/y=c/z时等好成立,
又25×36=30^2,所以
a/x=b/y=c/z=k
k^2=a^2/x^2=b^2/y^2=c^2/z^2=(a^2+b^2+c^2)/(x^2+y^2+z^2)=25/36 (合分比定理)
所以(2007a+5b+8c)/(2007x+5y+8z)=k=5/6