Question

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

Answer 1

Let the slopes of two lines are $m_1$ and $m_2$

Here $m_2=2m_1$

We know that the angle between them is $\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$

Here $\tan \theta =\frac{1}{3}$ and $m_2=2m_1$

$\frac{1}{3}=\left|\frac{m_1-2m_1}{1+m_1\times 2m_1}\right|$

$\frac{1}{3}=\left|\frac{-m_1}{1+2m_1^2}\right|$

$\frac{m_1}{1+2m_1^2}=\frac{1}{3}$

$3m_1=1+2m_1^2$

$2m_1^2-3m_1+1=0$

On solving we get

$m_1=1,\frac{1}{2}$

and $m_2=2,1$