Question

The slope of a line is double of the slope of another line. If tangent of the angle between them is $\frac{1}{3}$, find the slopes of the lines.

Answer 1

Let $m_1$ and $m$ be the slopes of two given line such that $m_1$$=2m$.

We know that if $\theta$ is the angle between the lines $l_1$ and $l_2$ with slopes $m_1$ and $m_2$,

then $\tan \theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|$

It is given that the tangent of the angle between the two lines is $\frac{1}{3}$

$\therefore \frac{1}{3}=\left|\frac{m-2m}{1+\left(2m\right)m}\right|$

$\Rightarrow \frac{1}{3}=\left|\frac{-m}{1+2m^2}\right|$

$\Rightarrow \frac{1}{3}=\frac{-m}{1+2m^2}$ or $\frac{1}{3}=-\left\{\frac{-m}{1+2m^2}\right\}=\frac{m}{1+2m^2}$

$\Rightarrow 2m^2+3m+1=0$ or $2m^2-3m+1=0$

$\Rightarrow m=-1,-\frac{1}{2}$ or $m=1,\frac{1}{2}$

Corresponding $m_1=-2,-1,2,1$