Question

The equations of $L_1$ and $L_2$ are y = px and y = qx respectively. Suppose $L_1$ makes twice as large of an angle with the horizontal (measured counter clockwise 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 product (pq) is ___

Answer 1

Let $p=tan2\theta$ and $q=tan\theta$ and p = 4q and $p\ne 0\therefore tan2\theta \ne 0so\theta \ne 0$
$tan2\theta =4tan\theta$
$\Rightarrow \frac{2tan\theta }{1-ta{n}^2\theta }=4tan\theta \Rightarrow 1=2-2ta{n}^2\theta \Rightarrow ta{n}^2\theta =\frac{1}{2}\therefore pq=tan2\theta .tan\theta =\frac{2ta{n}^2\theta }{1-ta{n}^2\theta }=\frac{1}{1-\left(\frac{1}{2}\right)}=2$