Question

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

Answer 1

$\text{Let slope of one line is m, then slope of another line is 2 m.}\text{Given, the tangent of the angle between them is}tan\theta =\frac{1}{3}\therefore \left|\frac{m_1-m_2}{1+m_2{m}_2}\right|=\frac{1}{3}\Rightarrow \frac{1}{3}=\left|\frac{m-2m}{1+m\times 2m}\right|\Rightarrow \frac{1}{3}=\left|\frac{-m}{1+2m^2}\right|\Rightarrow (1+2m^2)=3m\Rightarrow 2m^2-3m+1=0\text{Factorize it by splitting the middle term}\Rightarrow 2m^2-2m-m+1=0\Rightarrow 2m(m-1)-1(m-1)=0\Rightarrow (2m-1)(m-1)=02m-1=0orm-1=0\Rightarrow m=\frac{1}{2},m=1$