1952 AMC 12 Problem 23

Attempt Problem 23 of the 1952 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 1952 AMC 12 solutions, or check the answer key.

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23.

\dfrac{x^2-bx}{ax-c}=\dfrac{m-1}{m+1}

has roots which are numerically equal but of opposite signs, the value of

m

must be:

\dfrac{a-b}{a+b}

\dfrac{a+b}{a-b}

c

\dfrac1c

1

Clear the denominator and collect the equation as a quadratic in

x

Opposite roots have sum zero, so the coefficient of

x

must vanish

Cross-multiplication gives

\begin{aligned} (m+1)x^2 &-(m+1)bx\\ &-(m-1)ax\\ &\quad{}+(m-1)c=0. \end{aligned}

Roots that are equal in magnitude and opposite in sign have sum

0,

so the coefficient of

x

0.

Thus

m(a+b)+(b-a)=0,

and

m=\frac{a-b}{a+b}.

Thus, the correct answer is A.

Problem 23 in other years