Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
Ta có:
\(\frac{x^2+y^2+z^2}{a^2+b^2+c^2}=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\)
\(\Leftrightarrow\left(\frac{x^2}{a^2}-\frac{x^2}{a^2+b^2+c^2}\right)+\left(\frac{y^2}{b^2}-\frac{y^2}{a^2+b^2+c^2}\right)+\left(\frac{z^2}{c^2}-\frac{z^2}{a^2+b^2+c^2}\right)=0\)
\(\Leftrightarrow x^2.\frac{b^2+c^2}{a^2+b^2+c^2}+y^2.\frac{a^2+c^2}{a^2+b^2+c^2}+z^2.\frac{a^2+b^2}{a^2+b^2+c^2}=0\)
Vì a, b, c khác 0 nên dấu bằng xảy ra khi \(x=y=z=0\)
\(\Rightarrow M=x^{2016}+y^{2016}+z^{2016}=0^{2016}+0^{2016}+0^{2016}=0\)
\(\frac{x^2+y^2+z^2}{a^2+b^2+c^2}=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\)
\(\frac{x^2}{a^2+b^2+c^2}-\frac{x^2}{a^2}+\frac{y^2}{a^2+b^2+c^2}-\frac{y^2}{b^2}\)\(+\frac{z^2}{a^2+b^2+c^2}-\frac{z^2}{c^2}=0\)
\(x^2\left(\frac{1}{a^2+b^2+c^2}-\frac{1}{a^2}\right)\)\(+y^2\left(\frac{1}{a^2+b^2+c^2}-\frac{1}{b^2}\right)+z^2\left(\frac{1}{a^2+b^2+c^2}-\frac{1}{c^2}\right)\)\(=0\)
Vì \(\frac{1}{a^2+b^2+c^2}-\frac{1}{a^2}\ne0,\frac{1}{a^2+b^2+c^2}-\frac{1}{b^2}\ne0\)\(,\frac{1}{a^2+b^2+c^2}-\frac{1}{c^2}\ne0\) và \(a,b,c\ne0\)
\(\Rightarrow\left\{{}\begin{matrix}x^2=0\\y^2=0\\z^2=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=0\\y=0\\z=0\end{matrix}\right.\)\(\Rightarrow T=0\)
Ta có: \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\Leftrightarrow\left(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\right)^2=1\)
\(\Leftrightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\left(\frac{xy}{ab}+\frac{yz}{bc}+\frac{zx}{ca}\right)=1\)
\(\Leftrightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\cdot\frac{xyc+yza+zxb}{abc}=1\)
Mà \(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=0\Leftrightarrow\frac{yza+zxb+xyc}{xyz}=0\)
\(\Rightarrow yza+zxb+xyc=0\)
\(\Rightarrow A=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\)
\(\frac{x^2+y^2+z^2}{a^2+b^2+c^2}=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\)
\(\Leftrightarrow\left(\frac{x^2}{a^2}-\frac{x^2}{a^2+b^2+c^2}\right)+\left(\frac{y^2}{b^2}-\frac{y^2}{a^2+b^2+c^2}\right)+\left(\frac{z^2}{c^2}-\frac{z^2}{a^2+b^2+c^2}\right)=0\)
\(\Leftrightarrow\left(x^2.\frac{b^2+c^2}{a^2+b^2+c^2}\right)+\left(y^2.\frac{a^2+c^2}{a^2+b^2+c^2}\right)+\left(z^2.\frac{a^2+b^2}{a^2+b^2+c^2}\right)=0\)
Vì a,b,c khác
=>Dấu bằng xảy ra khi x=y=z=0
\(\Rightarrow x^{2014}+y^{2015}+z^{2016}=0^{2014}+0^{2015}+0^{2016}=0\)
Ta có \(\frac{x^2+y^2+z^2}{a^2+b^2+c^2}=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\)
\(\Leftrightarrow\frac{x^2}{a^2+b^2+c^2}+\frac{y^2}{a^2+b^2+c^2}+\frac{z^2}{a^2+b^2+c^2}=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\)
\(\Leftrightarrow\frac{x^2}{a^2+b^2+c^2}-\frac{x^2}{a^2}+\frac{y^2}{a^2+b^2+c^2}-\frac{y^2}{b^2}+\frac{z^2}{a^2+b^2+c^2}-\frac{z^2}{c^2}=0\)
\(\Leftrightarrow x^2\left(\frac{1}{a^2+b^2+c^2}-\frac{1}{a^2}\right)+y^2\left(\frac{1}{a^2+b^2+c^2}-\frac{1}{b^2}\right)+z^2\left(\frac{1}{a^2+b^2+c^2}-\frac{1}{c^2}\right)=0\)
Do \(\left\{\begin{matrix}\frac{1}{a^2+b^2+c^2}-\frac{1}{a^2}\\\frac{1}{a^2+b^2+c^2}-\frac{1}{b^2}\\\frac{1}{a^2+b^2+c^2}-\frac{1}{c^2}\end{matrix}\right.\ne0\) và \(a,b,c\ne0\)
\(\Rightarrow\left\{\begin{matrix}x^2=0\\y^2=0\\z^2=0\end{matrix}\right.\)
\(\Rightarrow\left\{\begin{matrix}x=0\\y=0\\z=0\end{matrix}\right.\)
Ta có \(A=x^{2008}+y^{2008}+z^{2008}\)
\(\Rightarrow A=0+0+0\)
\(\Rightarrow A=0\)
Vậy A = 0
Ta có
\(1\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=\frac{x^2+y^2+z^2}{a^2+b^2+c^2}\)
\(1\Leftrightarrow x^2+\frac{\left(b^2+c^2\right)x^2}{a^2}+y^2+\frac{\left(a^2+c^2\right)y^2}{b^2}+z^2+\frac{\left(a^2+b^2\right)z^2}{c^2}=x^2+y^2+z^2\)
\(\Leftrightarrow\frac{\left(b^2+c^2\right)x^2}{a^2}+\frac{\left(c^2+a^2\right)y^2}{b^2}+\frac{\left(a^2+b^2\right)z^2}{c^2}=0\)
Ta thấy rằng cả 3 phân số đó đều \(\ge0\)nên tổng 3 phân số sẽ \(\ge0\)
Dấu = xảy ra khi x = y = z = 0
Với x = y = z = 0 thì
\(\frac{x^{2016}}{a^{2016}}+\frac{y^{2016}}{b^{2016}}+\frac{z^{2016}}{c^{2016}}=\frac{x^{2016}+y^{2016}+z^{2016}}{a^{2016}+b^{2016}+c^{2016}}\Leftrightarrow\frac{0}{a^{2016}}+\frac{0}{b^{2016}}+\frac{0}{c^{2016}}=\frac{0+0+0}{a^{2016}+b^{2016}+c^{2016}}\)
\(\Leftrightarrow0=0\)(đúng)
\(\Rightarrow\)ĐPCM
a)Ta có: ab+ac+bc=-7 (ab+ac+bc)^2=49
nên
(ab)^2+(bc)^2+(ac)^2=49
nên a^4+b^4+c^4=(a^2+b^2+c^2)^2−2(ab)^2−2(ac)^2−2(bc^)2=98
b) (x^2+y^2+z^2)/(a^2+b^2+c^2)=
=x^2/a^2+y^2/b^2+z^2/c^2 <=>
x^2+y^2+z^2=x^2+(a^2/b^2)y^2+
+(a^2/c^2)z^2+(b^2/a^2)x^2+y^2+
+(b^2/c^2)z^2+(c^2/a^2)x^2+
+(c^2/b^2)y^2+z^2 <=>
[(b^2+c^2)/a^2]x^2+[(a^2+c^2)/b^2]y^2+
+[(a^2+b^2)/c^2]z^2 = 0 (*)
Đặt A=[(b^2+c^2)/a^2]x^2; B=[(a^2+c^2)/b^2]y^2;
và C=[(a^2+b^2)/c^2]z^2
Vì a,b,c khác 0 nên suy ra A,B,C đều không âm
Từ (*) ta có A+B+C=0
Tổng 3 số không âm bằng 0 thì cả 3 số đều phải bằng 0,tức A=B=C=0
Vì a,b,c khác 0 nên [(b^2+c^2)/c^2]>0 =>x^2=0 =>x=0
Tương tự B=C=0 =>y^2=z^2=0 => y=z=0
Vậy x^2011+y^2011+z^2011=0
Và x^2008+y^2008+z^2008=0.
Cô ơi em có cách khác ạ :)
\(\frac{x^2+y^2+z^2}{a^2+b^2+c^2}=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\)
\(\Leftrightarrow x^2\left(\frac{1}{a^2}-\frac{1}{a^2+b^2+c^2}\right)+y^2\left(\frac{1}{b^2}-\frac{1}{a^2+b^2+c^2}\right)+z^2\left(\frac{1}{c^2}-\frac{1}{a^2+b^2+c^2}\right)=0\)
Dấu "=" xảy ra tại x=y=z=0
Khi đó T=0
Ta có:
\(\frac{x^2+y^2+z^2}{a^2+b^2+c^2}=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\)
<=> \(\left(a^2+b^2+c^2\right)\)\(\frac{x^2+y^2+z^2}{a^2+b^2+c^2}=\left(a^2+b^2+c^2\right)\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\right)\)
<=> \(x^2+y^2+z^2=\left(a^2+b^2+c^2\right)\frac{x^2}{a^2}+\left(a^2+b^2+c^2\right)\frac{y^2}{b^2}+\left(a^2+b^2+c^2\right)\frac{z^2}{c^2}\)
<=> \(\frac{\left(b^2+c^2\right)}{a^2}x^2+\frac{\left(a^2+c^2\right)}{b^2}y^2+\frac{\left(a^2+b^2\right)}{c^2}z^2=0\)
vì a, b , c khác 0 nên \(\frac{\left(b^2+c^2\right)}{a^2};\frac{\left(c^2+a^2\right)}{b^2};\frac{\left(b^2+a^2\right)}{c^2}\ne0\)
\(\frac{\left(b^2+c^2\right)}{a^2}x^2\ge0;\frac{\left(a^2+c^2\right)}{b^2}y^2\ge0;\frac{\left(a^2+b^2\right)}{c^2}z^2\ge0\)với mọi x, y, z
=> \(\frac{\left(b^2+c^2\right)}{a^2}x^2+\frac{\left(a^2+c^2\right)}{b^2}y^2+\frac{\left(a^2+b^2\right)}{c^2}z^2\ge0\)với mọi x; y; z
Do đó: \(\frac{\left(b^2+c^2\right)}{a^2}x^2+\frac{\left(a^2+c^2\right)}{b^2}y^2+\frac{\left(a^2+b^2\right)}{c^2}z^2=0\)
=> x = y = z = 0
Vậy T = 0