Question

已知 $x$ 和 $y$ 是不同的非零实数，且 $x + \dfrac{2}{x} = y + \dfrac{2}{y}$ ， $xy$ 是多少？

Given that $x$ and $y$ are distinct nonzero real numbers such that $x + \dfrac{2}{x} = y + \dfrac{2}{y},$ what is $xy?$

$\dfrac{1}{4}$

$\dfrac{1}{2}$

$1$

$2$

$4$

两边同乘 $xy$ 来消去分母

Multiply both sides by $xy$ to clear the fractions

结果可因式分解为 $(x - y)(xy - 2) = 0$

The result factors as $(x - y)(xy - 2) = 0$

同乘 $xy$ 得 $x^2 y + 2y = xy^2 + 2x$ ，所以 $\begin{gathered} x^2 y - xy^2 - 2x + 2y \\ = (x - y)(xy - 2) \\ = 0 \end{gathered}\text{。}$

因为 $x \ne y$ ，所以 $xy = 2$ 。

因此，正确答案是 D。

Multiplying by $xy$ gives $x^2 y + 2y = xy^2 + 2x,$ so $\begin{gathered} x^2 y - xy^2 - 2x + 2y \\ = (x - y)(xy - 2) \\ = 0. \end{gathered}$

Since $x \ne y,$ it follows that $xy = 2.$

Thus, the correct answer is D.