\(P=\frac{x}{y}+\frac{y}{x}+\frac{xy}{x^2+y^2}\)
x,y>0
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\(a,ĐKXĐ:x\ne-;y\ne0\)
\(P=\frac{2}{x}-\left(\frac{x^2}{x^2+xy}+\frac{y^2-x^2}{xy}-\frac{y^2}{xy+y^2}\right)\cdot\frac{x+y}{x^2+xy+y^2}\)
\(P=\frac{2}{x}-\left(\frac{x^2}{x\left(x+y\right)}+\frac{y^2-x^2}{xy}-\frac{y^2}{y\left(x+y\right)}\right)\cdot\frac{x+y}{x^2+xy+y^2}\)
\(P=\frac{2}{x}-\left(\frac{x^2y}{xy\left(x+y\right)}+\frac{\left(x+y\right)\left(y^2-x^2\right)}{xy\left(x+y\right)}-\frac{xy^2}{xy\left(x+y\right)}\right)\cdot\frac{x+y}{x^2+xy+y^2}\)
\(P=\frac{2}{x}-\left(\frac{x^2y+xy^2-x^3+y^3-x^2y-xy^2}{xy\left(x+y\right)}\right)\cdot\frac{x+y}{x^2+xy+y^2}\)
\(P=\frac{2}{x}+\frac{x^3-y^3}{xy\left(x+y\right)}\cdot\frac{x+y}{x^2+xy+y^2}\)
\(P=\frac{2}{x}-\frac{\left(x-y\right)\left(x^2+xy+y^2\right)}{xy}\cdot\frac{1}{x^2+xy+y^2}\)
\(P=\frac{2}{x}-\frac{x-y}{xy}=\frac{2y-x+y}{xy}=\frac{3y-x}{xy}\)
\(b,x^2+y^2+10=2\left(x-3y\right)\)
\(\Leftrightarrow x^2+y^2+10=2x-6y\)
\(\Leftrightarrow x^2-2x+1+y^2+6y+9=0\)
\(\Leftrightarrow\left(x-1\right)^2+\left(y+3\right)^2=0\)
\(\Leftrightarrow\hept{\begin{cases}x=1\\y=-3\end{cases}}\)
thay vào P được : \(P=\frac{3\left(-3\right)-1}{-3\cdot1}=\frac{-10}{-3}=\frac{10}{3}\)
a, Rút gọn A
b,Tìm giá trị P, biết x,y thỏa mãn đẳng thức
x^2+y^2+10=2(x-3y)
Với đk trên ta có:
P = \(\frac{2}{x}-\left(\frac{x^2}{x^2+xy}+\frac{y^2-x^2}{xy}-\frac{y^2}{xy+y^2}\right).\frac{x+y}{x^2+xy+y^2}\)
\(=\frac{2}{x}-\left(\frac{x}{x+y}-\frac{\left(x-y\right)\left(x+y\right)}{xy}-\frac{y}{x+y}\right).\frac{x+y}{x^2+xy+y^2}\)
\(=\frac{2}{x}-\left(\frac{x-y}{x+y}-\frac{\left(x-y\right)\left(x+y\right)}{xy}\right).\frac{x+y}{x^2+xy+y^2}\)
\(=\frac{2}{x}-\frac{x-y}{xy}.\left(xy-\left(x+y\right)^2\right).\frac{1}{x^2+xy+y^2}\)
\(=\frac{2}{x}+\frac{x-y}{xy}\)
\(=\frac{x+y}{xy}\)
\(B=\left(\frac{x}{y^2+xy}-\frac{x-y}{x^2+xy}\right):\left(\frac{y^2}{x^3-xy^2}+\frac{1}{x+y}\right):\frac{3y}{y}\)
\(=\frac{x^2-xy+y^2}{xy\left(x+y\right)}\cdot\frac{x\left(x^2-y^2\right)}{x^2-xy+y^2}\cdot\frac{y}{3x}\)\(=\frac{x-y}{y}\cdot\frac{y}{3x}=\frac{x-y}{3x}\)
\(\frac{2}{x}-\left(\frac{x^2}{x^2+xy}-\frac{x^2-y^2}{xy}-\frac{y^2}{xy+y^2}\right)\)\(\left(\frac{x+y}{x^2+xy+y^2}\right)\)
ĐK: \(\hept{\begin{cases}x,y\ne0\\x\ne-y\end{cases}}\)
\(A=\frac{2}{x}-\frac{x^2y-\left(x-y\right)\left(x+y\right)^2-xy^2}{xy\left(x+y\right)}.\frac{x+y}{x^2+xy+y^2}\)
\(A=\frac{2}{x}+\frac{x^3-y^3}{xy\left(x+y\right)}.\frac{x+y}{x^2+xy+y^2}\)
\(A=\frac{2}{x}+\frac{x-y}{xy}\)
\(A=\frac{2y+x-y}{xy}\)
\(A=\frac{x+y}{xy}\)
\(A=\left[\frac{x^2-y^2}{xy}-\frac{1}{xy}\left(\frac{x^2}{y}-\frac{y^2}{x}\right)\right]:\frac{x-y}{xy}\)
\(A=\left[\frac{x^2-y^2}{xy}-\left(\frac{x}{y^2}-\frac{y}{x^2}\right)\right].\frac{xy}{x-y}\) => \(A=\left(\frac{x^2-y^2}{xy}-\frac{x^3-y^3}{x^2y^2}\right).\frac{xy}{x-y}=\left(\frac{\left(x-y\right)\left(x+y\right)}{xy}-\frac{\left(x-y\right)\left(x^2+xy+y^2\right)}{x^2y^2}\right).\frac{xy}{x-y}\)
=> \(A=\frac{x-y}{xy}\left(\left(x+y\right)-\frac{x^2+xy+y^2}{xy}\right).\frac{xy}{x-y}\)=> \(A=x+y-\frac{x^2+xy+y^2}{xy}=\frac{x^2y+xy^2-x^2-xy-y^2}{xy}\)