Question

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

Answer 1

Angle between lines with slope $m_1$ and $m_2$ is given by

$tan\theta =\left|\right(\frac{m_2-m_1}{1+m_1{m}_2}\left)\right|inancase\theta =\left(\frac{\pi }{4}\right),m_2=\left(\frac{1}{2}\right)\therefore tan\left(\frac{\pi }{4}\right)=\left|\right(\frac{\left(\frac{1}{2}\right)-m_1}{1+m_1\times \left(\frac{1}{2}\right)}\left)\right|1=\left|\right(\frac{1-2m_1}{2+m_1}\left)\right|\therefore \left(\frac{1-2m_1}{2+m_1}\right)=+1or1-2m_1=2+m_1\therefore 3m+1=-1or{m}_1=\left(\frac{1}{3}\right)also\left(\frac{1-2m_1}{2+m_1}\right)=-1or1-2m_1=-2-m_{1}or3=m_1\therefore m_1=3or\left(\frac{1}{3}\right)$