1966 AMC 12 Problem 33

Question

If

ab\ne0

and

|a|\ne|b|,

the number of distinct values of

x

satisfying the equation

\begin{aligned} \frac{x-a}{b}+\frac{x-b}{a} &=\frac{b}{x-a}\\ &\quad+\frac{a}{x-b} \end{aligned}

is:

0

1

2

3

4

Both sides contain the numerator

(a+b)x-(a^2+b^2)

after combining terms

Factor the equation into that numerator times a difference of reciprocals

Let

N=(a+b)x-a^2-b^2.

Combining each side gives

N\left(\frac1{ab} -\frac1{(x-a)(x-b)}\right)=0.

The first factor gives

x=\frac{a^2+b^2}{a+b}.

The second gives

(x-a)(x-b)=ab,

or

x(x-a-b)=0,

giving

x=0

and

x=a+b.

The hypotheses ensure that all three are defined and distinct.

Therefore, the correct answer is D.

Problem 33 in other years