1980 AHSME Problems/Problem 12

Problem

The equations of $L_1$ and $L_2$ are $y=mx$ and $y=nx$ , respectively. Suppose $L_1$ makes twice as large of 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 $L_2$ . If $L_1$ is not horizontal, then $mn$ is

$\text{(A)} \ \frac{\sqrt{2}}{2} \qquad \text{(B)} \ -\frac{\sqrt{2}}{2} \qquad \text{(C)} \ 2 \qquad \text{(D)} \ -2 \qquad \text{(E)} \ \text{not uniquely determined}$

Solution

Solution by e_power_pi_times_i

$4n = m$ , as stated in the question. In the line $L_1$ , draw a triangle with the coordinates $(0,0)$ , $(1,0)$ , and $(1,m)$ . Then $m = \tan(\theta_1)$ . Similarly, $n = \tan(\theta_2)$ . Since $4n = m$ and $\theta_1 = 2\theta_2$ , $\tan(2\theta_2) = 4\tan(\theta_2)$ . Using the angle addition formula for tangents, $\dfrac{2\tan(\theta_2)}{1-\tan^2(\theta_2)} = 4\tan(\theta_2)$ . Solving, we have $\tan(\theta_2) = 0, \dfrac{\sqrt{2}}{2}$ . But line $L_1$ is not horizontal, so therefore $(m,n) = (2\sqrt{2},\dfrac{\sqrt{2}}{2})$ . Looking at the answer choices, it seems the answer is $(2\sqrt{2})(\dfrac{\sqrt{2}}{2}) = \boxed{\text{(C)} \ 2}$ .

1980 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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1980 AHSME Problems/Problem 12

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Solution

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