Question

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 the value of the product(mn) equals.

• A. $\frac{\sqrt{2}}{2}$
• B. $-\frac{\sqrt{2}}{2}$
• C. $2$
• D. $-2$

Answer 1

The correct option is C $2$

Given lines $L_1:y=mx$

$L_2:y=nx$

$n=\tan \theta ,m=\tan 2\theta ,m=4n$

$m=\frac{2\tan \theta }{1-{\tan }^2\theta }$

$m=\frac{2n}{1-n^2}$

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

$2=\frac{1}{1-n^2}$

$n=\pm \frac{1}{\sqrt{2}}$

$m=\frac{4}{\sqrt{2}}$

$mn=\frac{4}{\sqrt{2}}\times \frac{1}{\sqrt{2}}=2$