1980 AMC 12 Problem 12

Attempt Problem 12 of the 1980 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 1980 AMC 12 solutions, or check the answer key.

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12.

The equations of

L_1

and

L_2

are

y=mx

and

y=nx,

respectively. Suppose

L_1

makes twice as large an angle with the horizontal (measured counterclockwise from the positive

x

-axis) as does

L_2,

and that

L_1

has

4

times the slope of

L_2.

L_1

is not horizontal, then

mn

\frac{\sqrt2}{2}

-\frac{\sqrt2}{2}

2

-2

not uniquely determined by the given information

Let the angle of inclination of

L_2

\theta

Substitute

m=4n

into

\tan(2\theta)=\frac{2\tan\theta}{1-\tan^2\theta}

Let

\tan\theta=n.

Then

m=\tan(2\theta)

and also

m=4n.

Hence

4n=\frac{2n}{1-n^2}.

The nonhorizontal condition gives

n\ne0,

2(1-n^2)=1,

n^2=\frac{1}{2}.

Thus

mn=4n^2=2.

Therefore, the correct answer is C.

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