The expression (a)/(a+y)+(y)/(a-y) (y)/(a+y)-(a)/(a-y), a real, a≠0, has the value -1 for:

(A) all but two real values of y

1963 AMC 12 Problem 17

Attempt Problem 17 of the 1963 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1963 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

17.

The expression

\frac{\dfrac{a}{a+y}+\dfrac{y}{a-y}} {\dfrac{y}{a+y}-\dfrac{a}{a-y}},

a

real,

a\ne0,

has the value

-1

for:

all but two real values of

y

only two real values of

y

all real values of

y

only one real value of

y

no real values of

y

Combine the two fractions in the numerator and denominator separately

Keep track of the excluded values

y=\pm a

For

y\ne\pm a,

the numerator simplifies to

\frac{a^2+y^2}{a^2-y^2}

and the denominator to its negative. Their quotient is therefore

-1.

The two values

y=a

and

y=-a

are undefined.

Thus, the correct answer is A.