Cho a+b+c=2 và a^2+b^2+c^2=4 và x/a=y/b=z/c Tính xy+yz+xz
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\(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}=t\Rightarrow\hept{\begin{cases}x=at\\y=bt\\z=ct\end{cases}}\).
\(4=\left(a+b+c\right)^2=a^2+b^2+c^2+2\left(ab+bc+ca\right)=4+2\left(ab+bc+ca\right)\)
\(\Rightarrow ab+bc+ca=0\)
\(P=xy+yz+zx=abt^2+bct^2+cat^2=t^2\left(ab+bc+ca\right)=0\)
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ta có: \(x+y+z=a\Rightarrow x^2+y^2+z^2+2\left(xy+yz+xz\right)=a^2\)
\(\Rightarrow b+2\left(xy+yz+xz\right)=a^2\Rightarrow xy+yz+xz=\frac{a^2-b}{2}\)
\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{c}\Rightarrow\frac{xy+yz+xz}{xyz}=\frac{1}{c}\Rightarrow c\left(xy+yz+xz\right)=xyz\)
Ta có:\(x^3+y^3+z^3=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)+3xyz\)
\(=a\left(b-\frac{a^2-b}{2}\right)+\frac{3c\left(a^2-b\right)}{2}\)
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ta có: \(\frac{x^2-yz}{a}=\frac{y^2-xz}{b}=\frac{z^2-xy}{c}\)
\(\Rightarrow\frac{a}{x^2-yz}=\frac{b}{y^2-xz}=\frac{c}{z^2-xy}\Rightarrow\frac{a^2}{\left(x^2-yz\right)^2}=\frac{b^2}{\left(y^2-xz\right)^2}=\frac{c^2}{\left(z^2-xy\right)^2}\) (1)
=> \(\frac{a}{\left(x^2-yz\right)}.\frac{a}{\left(x^2-yz\right)}=\frac{b}{y^2-xz}.\frac{c}{z^2-xy}=\frac{a^2}{\left(x^2-yz\right)^2}=\frac{bc}{\left(y^2-xz\right).\left(z^2-xy\right)}\)
a^2/(x^2-yz)^2 = (a^2-bc)/[(x^2-yz)^2 - (y^2-xz)(z^2-xy)] = (a^2-bc)/[x (x^3 + y^3 + z^3 - 3xyz)] =>
(a^2-bc)/x = [a^2/(x^2 - yz)^2] * (x^3 + y^3 + z^3 - 3xyz) (2)
Thực hiện tương tự ta cũng có
(b^2-ac)/y = [b^2/(y^2 - xz)^2] * (x^3 + y^3 + z^3 - 3xyz) (3)
(c^2-ab)/z = [c^2/(z^2 - xy)^2] * (x^3 + y^3 + z^3 - 3xyz) (4)
Từ (1),(2),(3),(4) => (a^2-bc)/x = (b^2-ac)/y = (c^2-ab)/z.
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Ta có : \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}==\frac{x+y+z}{a+b+c}=\frac{x+y+z}{1}\)
\(\frac{x^2}{a^2}=\frac{y^2}{b^2}=\frac{z^2}{c^2}=\frac{x^2+y^2+z^2}{a^2+b^2+c^2}=\frac{x^2+y^2+z^2}{1}\)
\(\left(x+y+z\right)^2=x^2+y^2+z^2\)
\(\Rightarrow2\left(xy+yz+zx\right)=0\)
\(\Rightarrow xy+yz+zx=0\)