A2. Let $a$ and $b$ be different real numbers. For which number $x$ does the equation $\frac{x-a}{x-b}=\frac{x-b}{x-a}$ hold?
(A) $\frac{a-b}{2}$
(B) $\frac{a^{2}+b^{2}}{a+b}$
(C) $\frac{a^{2}+b^{2}}{2(a+b)}$
(D) $a+b$
(E) $\frac{a+b}{2}$
A2. We multiply equation A2 by $(x-a)(x-b)$, to get $(x-a)^{2}=(x-b)^{2}$. After squaring, we subtract $x^{2}$ and get $-2 a x+a^{2}=-2 b x+b^{2}$, which can be rearranged to $2(b-a) x=$ $(b-a)(b+a)$. Since $b-a \neq 0$, we can divide the equation by $b-a$, to get $2 x=b+a$, from which we express $x=\frac{a+b}{2}$.