Cho x, y , x là các số thực thỏa mãn: \(\dfrac{x^4}{a}+\dfrac{y^4}{b}=\dfrac{x^2+y^2}{a+b};x^2+y^2=1\)
Chứng minh:\(\dfrac{x^{2006}}{a^{1003}}+\dfrac{y^{2006}}{b^{1003}}=\dfrac{2}{\left(a+b\right)^{1003}}\)
Phân thức đại số
Cho a,b,c và x,y,z là các số khác nhau và khác không chứng minh rằng nếu:
\(\dfrac{a}{x}+\dfrac{b}{y}+\dfrac{c}{z}=0\) và \(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1\) thì \(\dfrac{x^2}{a^2}+\dfrac{y^2}{z^2}+\dfrac{z^2}{c^2}=1\)
\(\dfrac{a}{x}+\dfrac{b}{y}+\dfrac{c}{z}=0\)
\(\Rightarrow\dfrac{abz}{xyz}+\dfrac{bxz}{xyz}+\dfrac{cxy}{xyz}=0\)
\(\Rightarrow\dfrac{abz+bxz+cxy}{xyz}=0\)
\(\Rightarrow abz+bxz+cxy=0\)
\(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1\)
\(\Rightarrow\left(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}\right)^2=1\)
\(\Rightarrow\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}+2\dfrac{xy}{ab}+2\dfrac{xz}{ac}+2\dfrac{yz}{bc}=1\)
\(\Rightarrow\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}+2\left(\dfrac{xy}{ab}+\dfrac{xz}{ac}+\dfrac{yz}{bc}\right)=1\)
\(\Rightarrow\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}+2\left(\dfrac{cxy}{abc}+\dfrac{bxz}{abc}+\dfrac{ayz}{abc}\right)=1\)
\(\Rightarrow\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}+2\left(\dfrac{cxy+bxz+ayz}{abc}\right)=0\)
\(\Rightarrow\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}+2.\left(\dfrac{0}{abc}\right)=1\)
\(\Rightarrow\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}+2.0=1\) \(\Rightarrow\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}+0=1\) \(\Rightarrow\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1\) ( đpcm )
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