Question

If the angle between two lines is $\frac{\pi }{4}$ and slope of one of lines is $\frac{1}{2}$. Find the slope of the other line.

Answer 1

If $\theta$ be the acute angle between the lines with slopes $m_{1}and{m}_2$ then $\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$

Let $\theta =\frac{\pi }{4}$ and $m_1=\frac{1}{2}$

$\therefore \tan \frac{\pi }{4}=\left|\frac{\frac{1}{2}-m_2}{1+\frac{1}{2}{m}_2}\right|$

$\Rightarrow 1=\left|\frac{1-2m_2}{2+m_2}\right|\Rightarrow \frac{1-2m_2}{2+m_2}=+1or-1$

Now $\frac{1-2m_2}{2+m_2}=1\Rightarrow m_2=-\frac{1}{3}and\frac{1-2m_2}{2+m_2}=-1\Rightarrow m_2=3$

$\therefore$ The slope of the other line is either -1/3 or 3