Question

两个非零实数 $a$ 和 $b$ 满足 $ab = a - b$ 求下面表达式的一个可能值： $\dfrac{a}{b} + \dfrac{b}{a} - ab.$

Two non-zero real numbers, $a$ and $b,$ satisfy $ab = a - b.$ Find a possible value of $\dfrac{a}{b} + \dfrac{b}{a} - ab.$

$-2$

$-\dfrac12$

$\dfrac13$

$\dfrac12$

$2$

通分到分母 $ab$ 得 $\dfrac{a}{b} + \dfrac{b}{a} - ab = \dfrac{a^2 + b^2 - (ab)^2}{ab}.$

代入 $ab = a - b$ ，得到 $\dfrac{a^2 + b^2 - (a - b)^2}{ab} = \dfrac{2ab}{ab} = 2.$

所以正确答案是 E。

Over the common denominator $ab,$ $\dfrac{a}{b} + \dfrac{b}{a} - ab = \dfrac{a^2 + b^2 - (ab)^2}{ab}.$

Substituting $ab = a - b$ gives $\dfrac{a^2 + b^2 - (a - b)^2}{ab} = \dfrac{2ab}{ab} = 2.$

Thus, the correct answer is E.