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.

• A. $3$ or $\frac{-1}{3}$
• B. $2$ or $\frac{-1}{2}$
• C. $4$ or $\frac{-1}{4}$
• D. $3$ or $-3$

Answer 1

Answer: (A)

$3$ or $\frac{-1}{3}$

Answer 2

We know that the acute angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by $tan\,\theta=\left|\frac{m_{2}-m_{1}}{1+m_{1}m_{2}}\right|\quad\ldots\left(i\right)$ Let $m_{1}=\frac{1}{2}$, $m_{2}=m$ and $\theta=\frac{\pi}{4}$. Now, putting these values in $\left(i\right)$, we get $tan \frac{\pi}{4}=\left|\frac{m-\frac{1}{2}}{1+\frac{1}{2}m}\right|$ $\Rightarrow 1=\left|\frac{2m-1}{2+m}\right|$, wihich gives $\frac{2m-1}{2+m}=1$ or $\frac{2m-1}{2+m}=-1$ Therefore $m = 3$ or $m=-\frac{1}{3}$ Hence, slope of the other line is $3$ or $-\frac{1}{3}$.